what is a convex body? … · what is a convex body? galynav.livshyts mathematical sciences...
TRANSCRIPT
![Page 1: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/1.jpg)
What is a convex body?
Galyna V. Livshyts
Mathematical Sciences Research Institute,Berkeley, CA
Field of dreams, Saint Louis, MONovember, 2017.
![Page 2: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/2.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.
We shall study geometry of Rn, an n-dimensional space.The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).For λ > 0, we have λx = (λx1, ...,λxn).Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 3: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/3.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.We shall study geometry of Rn, an n-dimensional space.
The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).For λ > 0, we have λx = (λx1, ...,λxn).Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 4: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/4.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.We shall study geometry of Rn, an n-dimensional space.The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).For λ > 0, we have λx = (λx1, ...,λxn).Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 5: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/5.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.We shall study geometry of Rn, an n-dimensional space.The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).
For λ > 0, we have λx = (λx1, ...,λxn).Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 6: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/6.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.We shall study geometry of Rn, an n-dimensional space.The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).For λ > 0, we have λx = (λx1, ...,λxn).
Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 7: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/7.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.We shall study geometry of Rn, an n-dimensional space.The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).For λ > 0, we have λx = (λx1, ...,λxn).Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 8: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/8.jpg)
n-dimensional Euclidean space
Throughout the talk, n shall stand for a positive integer, usually large.We shall study geometry of Rn, an n-dimensional space.The elements of Rn are vectors with n coordinates: x = (x1, ...,xn) ∈ Rn.
Given vectors x ,y ∈ Rn, with x = (x1, ...,xn) and y = (y1, ...,yn), their sumis the vector x + y = (x1 + y1, ...,xn + yn).For λ > 0, we have λx = (λx1, ...,λxn).Scalar product 〈x ,y〉= x1y1 + ...+ xnyn.
Galyna V. Livshyts What is a convex body?
![Page 9: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/9.jpg)
Convex bodies
Convex setA set K in Rn is called convex if for any pair of points x ,y ∈ K , the interval[x ,y ] is fully contained in K .
Equivalently, set K is convex if for every x ,y ∈ K and for every λ ∈ [0,1],the vector λx + (1−λ)y ∈ K .A convex body in Rn is a compact convex set with non-empty interior.A body K is called symmetric if x ∈ K =⇒ −x ∈ K .
Galyna V. Livshyts What is a convex body?
![Page 10: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/10.jpg)
Convex bodies
Convex setA set K in Rn is called convex if for any pair of points x ,y ∈ K , the interval[x ,y ] is fully contained in K .
Equivalently, set K is convex if for every x ,y ∈ K and for every λ ∈ [0,1],the vector λx + (1−λ)y ∈ K .
A convex body in Rn is a compact convex set with non-empty interior.A body K is called symmetric if x ∈ K =⇒ −x ∈ K .
Galyna V. Livshyts What is a convex body?
![Page 11: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/11.jpg)
Convex bodies
Convex setA set K in Rn is called convex if for any pair of points x ,y ∈ K , the interval[x ,y ] is fully contained in K .
Equivalently, set K is convex if for every x ,y ∈ K and for every λ ∈ [0,1],the vector λx + (1−λ)y ∈ K .A convex body in Rn is a compact convex set with non-empty interior.
A body K is called symmetric if x ∈ K =⇒ −x ∈ K .
Galyna V. Livshyts What is a convex body?
![Page 12: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/12.jpg)
Convex bodies
Convex setA set K in Rn is called convex if for any pair of points x ,y ∈ K , the interval[x ,y ] is fully contained in K .
Equivalently, set K is convex if for every x ,y ∈ K and for every λ ∈ [0,1],the vector λx + (1−λ)y ∈ K .A convex body in Rn is a compact convex set with non-empty interior.A body K is called symmetric if x ∈ K =⇒ −x ∈ K .
Galyna V. Livshyts What is a convex body?
![Page 13: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/13.jpg)
Convex bodies
Convex hull of N points x1, ...,xN
conv(x1, ...,xN) = {y ∈ Rn : y = λ1x1 + ...+λNxN , λi ≥ 0,∑
λi = 1}.
is a convex body:
An intersection of N half-spaces is a convex body:
Moreover, every convex body is an intersection of (very many) half-spaces,as well as the convex hull of its boundary points.
Galyna V. Livshyts What is a convex body?
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Convex bodies
Convex hull of N points x1, ...,xN
conv(x1, ...,xN) = {y ∈ Rn : y = λ1x1 + ...+λNxN , λi ≥ 0,∑
λi = 1}.
is a convex body:
An intersection of N half-spaces is a convex body:
Moreover, every convex body is an intersection of (very many) half-spaces,as well as the convex hull of its boundary points.
Galyna V. Livshyts What is a convex body?
![Page 15: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/15.jpg)
Convex bodies
Convex hull of N points x1, ...,xN
conv(x1, ...,xN) = {y ∈ Rn : y = λ1x1 + ...+λNxN , λi ≥ 0,∑
λi = 1}.
is a convex body:
An intersection of N half-spaces is a convex body:
Moreover, every convex body is an intersection of (very many) half-spaces,as well as the convex hull of its boundary points.
Galyna V. Livshyts What is a convex body?
![Page 16: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/16.jpg)
Examples of convex bodies
CUBEThe unit cube in Rn is the set of points
Bn∞ = [−1,1]n = {x ∈ Rn : |xi | ≤ 1, i = 1, ...,n}.
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
CUBEThe unit cube in Rn is the set of points
Bn∞ = [−1,1]n = {x ∈ Rn : |xi | ≤ 1, i = 1, ...,n}.
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
SIMPLEXConvex hull of n +1 points in Rn is called simplex:
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
SIMPLEXConvex hull of n +1 points in Rn is called simplex:
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
DIAMONDThe diamond, or cross-polytope is the set defined as
Bn1 = {x ∈ Rn : |x1|+ ...+ |xn| ≤ 1}.
In other words, Bn1 is the convex hull of points
(0,0, ...,1,0, ...,0),(0,0, ...,−1,0, ...,0), ...
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
DIAMONDThe diamond, or cross-polytope is the set defined as
Bn1 = {x ∈ Rn : |x1|+ ...+ |xn| ≤ 1}.
In other words, Bn1 is the convex hull of points
(0,0, ...,1,0, ...,0),(0,0, ...,−1,0, ...,0), ...
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
BALLThe unit ball in Rn is the set
Bn2 = {x ∈ Rn : |x1|2 + ...+ |xn|2 ≤ 1}.
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
BALLThe unit ball in Rn is the set
Bn2 = {x ∈ Rn : |x1|2 + ...+ |xn|2 ≤ 1}.
Galyna V. Livshyts What is a convex body?
![Page 24: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/24.jpg)
Examples of convex bodies
Lp-BALL
More generally, for p ≥ 1, Lp-ball in Rn is the set
Bnp = {x ∈ Rn : |x1|p + ...+ |xn|p ≤ 1}.
p = 2 – usual “euclidean” ball;p = 1 – cross-polytope;p =∞ – cube!
Galyna V. Livshyts What is a convex body?
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Examples of convex bodies
Lp-BALL
More generally, for p ≥ 1, Lp-ball in Rn is the set
Bnp = {x ∈ Rn : |x1|p + ...+ |xn|p ≤ 1}.
p = 2 – usual “euclidean” ball;p = 1 – cross-polytope;p =∞ – cube!
Galyna V. Livshyts What is a convex body?
![Page 26: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/26.jpg)
Ball, cube, diamond
Ball, cube and diamond are very far!Note that
1√n Ball ⊂ Diamond
⊂ Ball ⊂ Cube ⊂√
n ·Ball
Galyna V. Livshyts What is a convex body?
![Page 27: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/27.jpg)
Ball, cube, diamond
Ball, cube and diamond are very far!Note that
1√n Ball ⊂ Diamond ⊂ Ball
⊂ Cube ⊂√
n ·Ball
Galyna V. Livshyts What is a convex body?
![Page 28: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/28.jpg)
Ball, cube, diamond
Ball, cube and diamond are very far!Note that
1√n Ball ⊂ Diamond ⊂ Ball ⊂ Cube
⊂√
n ·Ball
Galyna V. Livshyts What is a convex body?
![Page 29: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/29.jpg)
Ball, cube, diamond
Ball, cube and diamond are very far!Note that
1√n Ball ⊂ Diamond ⊂ Ball ⊂ Cube ⊂
√n ·Ball
Galyna V. Livshyts What is a convex body?
![Page 30: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/30.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt =
Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 31: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/31.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 32: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/32.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt
≈∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 33: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/33.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt =
C√n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 34: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/34.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds =
C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 35: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/35.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
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Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 37: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/37.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!
The radius of the ball of unit volume is of order√
n!
Galyna V. Livshyts What is a convex body?
![Page 38: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/38.jpg)
Volumes of convex bodies
Volume of the unit ballBy Fubbini’s theorem,
Voln(Bn2 ) =
∫ 1
−1Voln−1(
√1− t2Bn−1
2 )dt = Voln−1(Bn−12 )
∫ 1
−1(1− t2)
n−12 dt;
Therefore,
Voln(Bn2 )
Voln−1(Bn−12 )
=∫ 1
−1(1− t2)
n−12 dt ≈
∫ 1
−1e−
nt22 dt = C√
n
∫e−s2
2 ds = C ′√n.
Voln(Bn2 ) =
(√2πe√n
)n;
The unit ball gets smaller and smaller!The radius of the ball of unit volume is of order
√n!
Galyna V. Livshyts What is a convex body?
![Page 39: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/39.jpg)
Hyperplanes and slabs
Hyperplanes
For a unit vector ξ, define the hyperplane H = ξ⊥, orthogonal to ξ, as
ξ⊥ = {x ∈ Rn : 〈x , ξ〉= 0}.
SlabsFor a unit vector ξ, define the slab Sξ, orthogonal to ξ, of width ρ, as
Sξ = {x ∈ Rn : |〈x , ξ〉| ≤ ρ}.
Galyna V. Livshyts What is a convex body?
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Hyperplanes and slabs
Hyperplanes
For a unit vector ξ, define the hyperplane H = ξ⊥, orthogonal to ξ, as
ξ⊥ = {x ∈ Rn : 〈x , ξ〉= 0}.
SlabsFor a unit vector ξ, define the slab Sξ, orthogonal to ξ, of width ρ, as
Sξ = {x ∈ Rn : |〈x , ξ〉| ≤ ρ}.
Galyna V. Livshyts What is a convex body?
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Distribution of mass in the ball
Note that Voln−1(Bn−12 ∩ξ⊥)
Voln(Bn2∩ξ⊥) = c
√n – large!
Let B be the ball of unit volume, i.e. c√
nBn2 .
Note that Voln−1(B∩ ξ⊥)≈√
e
Therefore, the constant portion of the volume of the ball is contained in aslab of constant width!
Galyna V. Livshyts What is a convex body?
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Distribution of mass in the ball
Note that Voln−1(Bn−12 ∩ξ⊥)
Voln(Bn2∩ξ⊥) = c
√n – large!
Let B be the ball of unit volume, i.e. c√
nBn2 .
Note that Voln−1(B∩ ξ⊥)≈√
e
Therefore, the constant portion of the volume of the ball is contained in aslab of constant width!
Galyna V. Livshyts What is a convex body?
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Distribution of mass in the ball
Note that Voln−1(Bn−12 ∩ξ⊥)
Voln(Bn2∩ξ⊥) = c
√n – large!
Let B be the ball of unit volume, i.e. c√
nBn2 .
Note that Voln−1(B∩ ξ⊥)≈√
e
Therefore, the constant portion of the volume of the ball is contained in aslab of constant width!
Galyna V. Livshyts What is a convex body?
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Distribution of mass in the ball
Note that Voln−1(Bn−12 ∩ξ⊥)
Voln(Bn2∩ξ⊥) = c
√n – large!
Let B be the ball of unit volume, i.e. c√
nBn2 .
Note that Voln−1(B∩ ξ⊥)≈√
e
Therefore, the constant portion of the volume of the ball is contained in aslab of constant width!
Galyna V. Livshyts What is a convex body?
![Page 45: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/45.jpg)
More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..
No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere. Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ = |S
n−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
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More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere. Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ = |S
n−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
![Page 47: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/47.jpg)
More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere.
Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ = |S
n−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
![Page 48: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/48.jpg)
More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere. Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ =
|Sn−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
![Page 49: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/49.jpg)
More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere. Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ = |S
n−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
![Page 50: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/50.jpg)
More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere. Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ = |S
n−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
![Page 51: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/51.jpg)
More on distribution of mass in the ball
We know that most of the mass of the ball comes from thin slabs. Does itmean that the volume is concentrated in the center?..No!
Most of the mass of the ball is near the boundary!
Denote the boundary of the unit ball by Sn−1 – unit sphere. Integrating inpolar coordinates, we get
Voln(Bn2 ) =
∫Sn−1
∫ 1
0tn−1dtdθ = |S
n−1|n .
Note that ∫ 1
0tn−1 = const ·
∫ 1
1− 1n
tndt.
Therefore, a constant portion of the volume of the unit ball is in the thinspherical shell near the boundary!
Galyna V. Livshyts What is a convex body?
![Page 52: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/52.jpg)
More on distribution of mass in the ball
Galyna V. Livshyts What is a convex body?
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Smallest section of the unit cube
Consider a unit (in volume) cube [− 12 ,
12 ]n ⊂ Rn.
What is its smallest section?
Turns out that the smallest (in area) section of the unit cube is the one parallelto coordinate subspaces:
Galyna V. Livshyts What is a convex body?
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Smallest section of the unit cube
Consider a unit (in volume) cube [− 12 ,
12 ]n ⊂ Rn. What is its smallest section?
Turns out that the smallest (in area) section of the unit cube is the one parallelto coordinate subspaces:
Galyna V. Livshyts What is a convex body?
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Smallest section of the unit cube
Consider a unit (in volume) cube [− 12 ,
12 ]n ⊂ Rn. What is its smallest section?
Turns out that the smallest (in area) section of the unit cube is the one parallelto coordinate subspaces:
Galyna V. Livshyts What is a convex body?
![Page 56: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/56.jpg)
Largest section of the unit cube
What is its largest section of the cube [− 12 ,
12 ]n ⊂ Rn?
Theorem (Keith Ball, 1984)
For every dimension n and for every unit vector u ∈ Rn,
Voln−1([−12 ,
12 ]n ∩u⊥)≤
√2.
This estimate is sharp!
Galyna V. Livshyts What is a convex body?
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Largest section of the unit cube
What is its largest section of the cube [− 12 ,
12 ]n ⊂ Rn?
Theorem (Keith Ball, 1984)
For every dimension n and for every unit vector u ∈ Rn,
Voln−1([−12 ,
12 ]n ∩u⊥)≤
√2.
This estimate is sharp!
Galyna V. Livshyts What is a convex body?
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Largest section of the unit cube
What is its largest section of the cube [− 12 ,
12 ]n ⊂ Rn?
Theorem (Keith Ball, 1984)
For every dimension n and for every unit vector u ∈ Rn,
Voln−1([−12 ,
12 ]n ∩u⊥)≤
√2.
This estimate is sharp!
Galyna V. Livshyts What is a convex body?
![Page 59: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/59.jpg)
Slicing problem
We have discovered that sections of the ball of unit volume are of area√
e.
The largest section of the cube of unit volume is of area√2.
They are both constants that do not depend on the dimension!
Bourgain’s slicing problem, 1982Consider a convex body K in Rn of volume one. How small can the area of itslargest hyperplane section be? Can it be smaller than an absolute constant?
Nobody knows the answer!This question is wide open! The best known estimate is due to Klartag, and itsays that the largest section of a convex body of unit volume cannot be smallerthan cn−
14 .
Galyna V. Livshyts What is a convex body?
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Slicing problem
We have discovered that sections of the ball of unit volume are of area√
e.The largest section of the cube of unit volume is of area
√2.
They are both constants that do not depend on the dimension!
Bourgain’s slicing problem, 1982Consider a convex body K in Rn of volume one. How small can the area of itslargest hyperplane section be? Can it be smaller than an absolute constant?
Nobody knows the answer!This question is wide open! The best known estimate is due to Klartag, and itsays that the largest section of a convex body of unit volume cannot be smallerthan cn−
14 .
Galyna V. Livshyts What is a convex body?
![Page 61: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/61.jpg)
Slicing problem
We have discovered that sections of the ball of unit volume are of area√
e.The largest section of the cube of unit volume is of area
√2.
They are both constants that do not depend on the dimension!
Bourgain’s slicing problem, 1982Consider a convex body K in Rn of volume one. How small can the area of itslargest hyperplane section be? Can it be smaller than an absolute constant?
Nobody knows the answer!This question is wide open! The best known estimate is due to Klartag, and itsays that the largest section of a convex body of unit volume cannot be smallerthan cn−
14 .
Galyna V. Livshyts What is a convex body?
![Page 62: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/62.jpg)
Slicing problem
We have discovered that sections of the ball of unit volume are of area√
e.The largest section of the cube of unit volume is of area
√2.
They are both constants that do not depend on the dimension!
Bourgain’s slicing problem, 1982Consider a convex body K in Rn of volume one. How small can the area of itslargest hyperplane section be? Can it be smaller than an absolute constant?
Nobody knows the answer!This question is wide open! The best known estimate is due to Klartag, and itsays that the largest section of a convex body of unit volume cannot be smallerthan cn−
14 .
Galyna V. Livshyts What is a convex body?
![Page 63: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/63.jpg)
Slicing problem
We have discovered that sections of the ball of unit volume are of area√
e.The largest section of the cube of unit volume is of area
√2.
They are both constants that do not depend on the dimension!
Bourgain’s slicing problem, 1982Consider a convex body K in Rn of volume one. How small can the area of itslargest hyperplane section be? Can it be smaller than an absolute constant?
Nobody knows the answer!This question is wide open!
The best known estimate is due to Klartag, and itsays that the largest section of a convex body of unit volume cannot be smallerthan cn−
14 .
Galyna V. Livshyts What is a convex body?
![Page 64: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/64.jpg)
Slicing problem
We have discovered that sections of the ball of unit volume are of area√
e.The largest section of the cube of unit volume is of area
√2.
They are both constants that do not depend on the dimension!
Bourgain’s slicing problem, 1982Consider a convex body K in Rn of volume one. How small can the area of itslargest hyperplane section be? Can it be smaller than an absolute constant?
Nobody knows the answer!This question is wide open! The best known estimate is due to Klartag, and itsays that the largest section of a convex body of unit volume cannot be smallerthan cn−
14 .
Galyna V. Livshyts What is a convex body?
![Page 65: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/65.jpg)
Central Limit Theorem and the cube
Central Limit TheoremLet X1, ...,Xn be independent identically distributed random variables. ThenX1+...+Xn√
n has almost normal (gaussian) distribution.
Example: uniform random variables on [0,1].Note that in the case when Xi are uniform on the unit interval,
X1 + ...+ Xn√n
= 〈X ,u〉,
where u = ( 1√n , ...,
1√n ) and X is a random vector uniformly distributed in
the unit cube.The density of 〈X ,u〉 takes values which are hyperplane sections of thecube, orthogonal to u!
Galyna V. Livshyts What is a convex body?
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Central Limit Theorem and the cube
Central Limit TheoremLet X1, ...,Xn be independent identically distributed random variables. ThenX1+...+Xn√
n has almost normal (gaussian) distribution.
Example: uniform random variables on [0,1].Note that in the case when Xi are uniform on the unit interval,
X1 + ...+ Xn√n
= 〈X ,u〉,
where u = ( 1√n , ...,
1√n ) and X is a random vector uniformly distributed in
the unit cube.
The density of 〈X ,u〉 takes values which are hyperplane sections of thecube, orthogonal to u!
Galyna V. Livshyts What is a convex body?
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Central Limit Theorem and the cube
Central Limit TheoremLet X1, ...,Xn be independent identically distributed random variables. ThenX1+...+Xn√
n has almost normal (gaussian) distribution.
Example: uniform random variables on [0,1].Note that in the case when Xi are uniform on the unit interval,
X1 + ...+ Xn√n
= 〈X ,u〉,
where u = ( 1√n , ...,
1√n ) and X is a random vector uniformly distributed in
the unit cube.The density of 〈X ,u〉 takes values which are hyperplane sections of thecube, orthogonal to u!
Galyna V. Livshyts What is a convex body?
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Central Limit Theorem and the cube
CLT and the cubeThe Central Limit Theorem tells us that the sections of the cube have almostGaussian distribution!
Galyna V. Livshyts What is a convex body?
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Central Limit Theorem for convex sets
Central Limit Theorem for convex sets (Klartag 2006)
Sections of an arbitrary convex body in Rn in most of directions have almostGaussian distribution!
Thus, all the convex bodies are a bit like cubes!Also, all the convex bodies are a bit like balls: for every convex body in Rn
there is a section (of small dimension) which looks very much like a ball!This is the content of Milman-Dvoretszki theorem.But we said that balls and cubes are very far from each other... Oops!
Galyna V. Livshyts What is a convex body?
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Central Limit Theorem for convex sets
Central Limit Theorem for convex sets (Klartag 2006)
Sections of an arbitrary convex body in Rn in most of directions have almostGaussian distribution!
Thus, all the convex bodies are a bit like cubes!
Also, all the convex bodies are a bit like balls: for every convex body in Rn
there is a section (of small dimension) which looks very much like a ball!This is the content of Milman-Dvoretszki theorem.But we said that balls and cubes are very far from each other... Oops!
Galyna V. Livshyts What is a convex body?
![Page 71: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/71.jpg)
Central Limit Theorem for convex sets
Central Limit Theorem for convex sets (Klartag 2006)
Sections of an arbitrary convex body in Rn in most of directions have almostGaussian distribution!
Thus, all the convex bodies are a bit like cubes!Also, all the convex bodies are a bit like balls: for every convex body in Rn
there is a section (of small dimension) which looks very much like a ball!This is the content of Milman-Dvoretszki theorem.
But we said that balls and cubes are very far from each other... Oops!
Galyna V. Livshyts What is a convex body?
![Page 72: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/72.jpg)
Central Limit Theorem for convex sets
Central Limit Theorem for convex sets (Klartag 2006)
Sections of an arbitrary convex body in Rn in most of directions have almostGaussian distribution!
Thus, all the convex bodies are a bit like cubes!Also, all the convex bodies are a bit like balls: for every convex body in Rn
there is a section (of small dimension) which looks very much like a ball!This is the content of Milman-Dvoretszki theorem.But we said that balls and cubes are very far from each other... Oops!
Galyna V. Livshyts What is a convex body?
![Page 73: What is a convex body? … · What is a convex body? GalynaV.Livshyts Mathematical Sciences Research Institute, Berkeley, CA Fieldofdreams,SaintLouis,MO November,2017](https://reader034.vdocuments.net/reader034/viewer/2022050213/5f5f6421c65336555d400713/html5/thumbnails/73.jpg)
Thanks for your attention!
Galyna V. Livshyts What is a convex body?