Journal of Geometry. Vol. 3/2, 1973. Birkh~user Verlag Basel

TARSKI'S DEFINITION OF POINT IN B~NACH SPACES

Theodore Sullivan

In [4] A. Tarski uses Le{niewski's logical system of

mereology to define the notion of point. This definition

is done using only the name solid and the primitive rela-

tion of mereology "A is part of (or equals) B" where A and

B are solids (see below). He then shows that when solid is

interpreted as "sphere" in Euclidean space, points turn

out to be equivalence classes of concentric spheres. In

this paper we show that if Euclidean space is replaced by

Banach space (2 dimensional) then points turn out to be

equivalence classes of concentric open balls (i.e. concen-

tric is an equivalence relation) iff the space is strictly

convex. For an account of mereology the reader if referred

to [i] or [2]. For the elementary definitions and proper-

ties of Banach spaces the reader is referred to [5].

DEFINITIONS AND NOTATION

Tarski's definition of point is done in the following

six steps (A, B, C, X, and Y should be interpreted as

(open) balls):

(I) A is externally tangent to B iff A is disjoint from B

and for all X and Y, if B is part of X and Y and both X

and Y are disjoint from A then X is part of Y or Y is part

of X.

(2) A is internally tangent to B iff A is part of B, A

is not B, and for all X and Y, if A is part of X and Y and

both X and Y are part of B then X is part of Y or Y is

part of X.

(3) A and B are externall Y diametrical to C iff A and B

are externally tangent to C and for all X and Y, if A is

part of X, B is part of Y, and both X and Y are disjoint

from C then X is disjoint from Y.

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2 SULLIVAN

(~) A and B are internally diametrical to C iff A and B

are internally tangent to C and for all X and Y, if X and

Y are disjoint from C, X is externally tangent to A, and

Y is externally tangent to B then X and Y are disjoint.

(5) A is concentric to B iff (i) A is B or (ii) A is part

of B and whenever X and Y are such that X and Y are exter-

nally diametrical to A and internally tangent to B then

X and Y are internally diametrical to B or (iii) substitute

B for A and A for B in (ii~.

(6) A point is an equivalence class of concentric balls.

We shall make use of the following definitions where

denotes an arbitrary Banach space:

A (open) ball is a set of points in ~ of the form

{x I fix - all < r} where the radius r is a strictly posi-

Tive real number and the center a is a point of ~ We

shall use capital letters for balls and whenever a capi-

tal letter (A) denoting a ball occurs in a lemma the

smaller case letter (a) will denote the center throughout

the given lemma. Further the unit ball denotes the set

{xl fix - < l}.

A point p is collinear with points u and v in ~ if

p = u + t(v - u) where t is some real number. If O<t<l

then p is between u and v.

is said to be strictly convex if for any two points

p and q on the boundary of the unit ball it is not the

case that a point v is both between p and q and on the

boundary of the unit ball.

A ball A is strictly convex at p if p is a boundary

point of A and no line through p intersects more than one

boundary point of A - {p}.

Given a ray h emanating from a ~ ~ , the ray h'

emanating from a such that h u h' is a line is called the

complementary ray to h.

Finally, we shall use the conventions ~ to denote

the line segment from p to q and Bd A to denote the

boundary of A.

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SULLIVAN 3

MAIN THEOREM

Our principle result is the following:

Theorem. If 8 is a Banach space whose dimension is equal

to two then the relation "A is concentric to B" is an

equivalence relation in ~ iff ~ is strictly convex.

BALLS IN 2 DIMENSIONAL B~NACH SPACES

In this section we prove three technical lemmas about

balls in a two dimensional Banach space. These lemmas

depend essentially on the following well known result:

LEMMA I. If C is a closed compact convex set in B and q is

in the interior of C then each half line starting at q

meets the boundary of C exactly once.

LEMMA 2. Let A be a sphere in ~ and let u belong to BdA.

Then there exists a supporting cone R at u (constructed

below) which has the following property: p is in the

interior of R iff there exists a ball B s.t. p E B, the

center b of B lies on the ray from u through a, and the

radius of B is llb - nil.

PROOF: The set R shall consist of the union of all rays

from u which pass through a point on the boundary of A. To

show the necessity let p belong to the interior of R. The

ball B is constructed as follows (see figure i); choose

a point q in the interior of R s.t. p is between a and q

and then let v be the unique point of Bd A s.t. v, q, and u

are collinear (uniqueness follows from the fact that if

there were a second point satisfying the conditions then,

using lemma I, the ray through v from u would be a boundary

ray contradicting q is an interior point). The center b of

B is then the point on the ray from u through a satisfying

b-q is parallel to v-a. The radius of B is llb - nil. To

show the sufficiency of the property let B be a ball s.t.

b lies on the ray from u through a. It is easy to see that

each point v E Bd B distinct from u determines a ray

through u which intersects Bd A. Thus v e R. It now

follows that each point of B belongs to the interior of R

since R is convex. This completes the proof of lemma 2.

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We remark that it is not difficult to show that R is

a cone which has no boundary ray iff A is strictly convex

at u. q

v

b u

Figure 1

For our last lemma of this section we need the

following:

DEFINITION: The complementary cone of supDort R' of R at

u g Bd A (A a ball) is the union of the rays which are

complementary to the rays in R.

LEMMA 3. Let u I and u 2 be two distinct points on Bd A where

A is a ball in ~ . Further assume (ul-a) : ~ (u2-a) and

that A is strictly convex at u I and u 2. Then the interior

of the intersection of the complementary cones of support

at u I and u- 2 resp. is not empty (see figure 2).

PROOF: We assume without loss of generality that a = 0 and

u I lies on the positive x axis. By the symmetry of A the

cone of support T at -u I is just a translation of the

complementary cone of support R' at u I along a line which

passes through the interior of R' (i.e. the x axis). Thus

any ray in T (except a boundary ray) will eventually be in

R'. Now consider the special ray h from -u I through u 2 and

let h I denote the ray from u 2 contained in h. Now the

complementary ray h i is contained in the cone of support

S for A at u 2. Thus h I is contained in S' Since A is

strictly convex at u I and u 2 it follows that A is strictly

convex at -u I and that R', S' , and T are open sets by the

remark following lemma 2. Thus since h is in T it is

eventually in R' which implies h I is eventually in R' n S'

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The fact that R' n S' is open completes the proof of this

lemma.

-u I

R'~ S'

u 2

a ]Ul

A

Figure 2

PROOF OF SUFFICIENCY

The proof consists of a sequence of lemmas which

essentially characterize in terms of points the interpre-

tation of the steps of Tarski's definition. Since this

process was carried out for Hilbert spaces in [3] we shall

merely indicate, after the word proof, the appropriate

lemma number in [3] whenever a different proof is not

needed in our strictly convex situation.

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LEMMA 4. Let u, v, and p be distinct points of ~ Then

II v - ull = II u - Pll + lip - vll iff p is between u and v.

PROOF: Follows easily from the definition of strictly

convex.

LEMMA 5. Let A and B be two balls in ~ . A and B are

externally tangent iff A and B are disjoint and there

exists a unique point p e Bd A n Bd B. We remark that p

will lie between a and b. We shall refer to p as the

point of tangency.

PROOF: Lemmas 2 and 3 of [3].

LEMMA 6. A is internally tangent to B iff A is a proper

subset of B and there is a p e ~ such that Bd A n Bd B =

{p}. PROOF: Proof is similar to the proof of lemma 5.

LEMMA 7. Let A and B be externally diametrical to C. Then

Pl-C =-(P2-C) where Pl and P2 are the points of tangency 6f

A with C and B with C respectively according to lemma 5.

PROOF: We need a new proof here. It is clear pl~P2 .

Assume now Pl,C~(P2-C). Then since ~ is strictly con-

vex it follows that C is strictly convex at Pl and at P2"

Thus the intersection S of the complementary cones of sup-

port for C at Pl and P2 has a non empty interior by lemma 3.

It is also easy to see from the similarity of balls and

the remarks following lemma 5 that the complementary cones

of support at Pl and P2 are the cones of support for A and

B resp. at Pl and P2" Applying now lemma 2 to a point in

the interior of S we have balls X and Y containing A and

B resp. disjoint from C such that X n Y # ~ which is a

contradiction. Thus (Pl-C) = -(P2-C) as was to be shown.

LEMMA 8. If A, B, and C satisfy A and B are each exter-

nally tangent to C and c is between a and b then A and B

are externally diametrical to C.

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SULLIVAN 7

PROOF: Lemma 7 in [3].

LEMMA 9. If A and B are internally tangent to C then A and

B are internally diametrical to C iff c is between a and b.

PROOF: Lamina 8 in [3].

LEMMA i0. If A and B are balls in ~ s.t. a ~ b then A

is not concentric to B.

PROOF: The proof of lemma 9 in [3] can be followed here

except for the existence of the point g (and g') (the point

d cannot be shown to exist in our more general space). To

get g and g' we define a real function f by f(t)=tIIq-el I +

Ile-a+t(q-e)I I. We have then f(0) < radius of A < f(1).

Since f is continuous there exists t0s.t. 0 < t O < 1 and

f(t 0) = radius of A. Let g = e + t0(q-e), g' is done

similarly using q' instead of q. The rest of the proof

carries over here.

With Lemma i0 we can now conclude our proof of suffi-

ciency as follows: It is easy to show that if two balls

do have the same center then they are concentric. Thus

we have shown that two spheres are concentric iff they

have the same center which is enough to establish that

concentric is an equivalence relation.

PROOF OF NECESSITY

Before going into the proof we need the following two

laminas :

LEMMA ii. If A is externally (internally) tangent to B in

then A and B are strictly convex at the point of tan-

gency given in lemma 5.

PROOF: If we assume the conclusion false we have intui-

tively a flat side of both balls at p which allows more

than one point of tangency by sliding the balls along

the flat side (figure 3). This contradicts lemma 5. A

more precise argument is given in the counter example at

the end of [3]. We remark there will always be one point

p E Bd A n Bd B between a and b if A is externally tangent

to B. Strictly convex says there is only one point of

tangency.

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LEMMA 12. If A and B are externally diametrical to C in

then c is between a and b.

PROOF: Use the proof of lemma 7 after first applying

lemma ii.

Figure 3

We now begin the proof of the necessity. The method of

proof will be to construct a ball A with the following

properties (see fig. 4):

(i) A is contained in the unit ball U.

(2) The relation "A is concentric to U" is satisfied.

(3) The center of A is not the center of U.

It then follows easily that there exist balls A' and U'

s.t. A' is concentric to A and U is concentric to U' but

A' and U' are chosen small enough to be disjoint. This vio-

lates the transitivity property of the concentric relation.

The ball A is constructed as follows (see fig. 4).

SinceSis not strictly convex there exist distinct vectors

d and e s.t. lldll = 1 = llell and for real t, 0 < t < i,

we have lid + t(e - d)l I = i. Next we choose t 0 s.t.

0 < t 0 < 1 and 1/2(1 - t 0) lie - dll - t 0 >t o (this is pos-

sible since the function f(t)=i/2(l-t) IIe-dll-2t is con-

tinuous with f(!) < 0 <f(0)). Finally we let the vector

a = 1/2(1 - t0)(d + e). Now the ball A is defined to be

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SULLIVAN

the set {x I llx - all < t0}.

d

e

Figure 4

Properties (I) and (3) are easily seen to be satisfied

by the ball A. To establish the second property we shall

show that if two balls B and ~ are externally diametrical

to A then it is not possible that both B and F are inter-

nally tangent to U. To this end let g = (i - t0)d and

h = (i - t0)e. We now note the following three geometric

facts which are easily shown to hold:

(i) Each line through a must intersect one of the

line segments: gd, de, or e--h.

(it) By lemma 12 the points of tangency w and v of B

and F with A satisfy a is between b and f.

(iii) For B and F of (it) and line Z through a and b,

let p be the point on ~ determined by (i). Then either w

lies between d and p or v lies between d and p.

We shall assume without loss of generality that w lies

between d and p and either p e ~ or p e ~. We shall show

that with these assumptions B cannot be internally tangent

to U. First assume that p=g. Since B is externally tangent

to A at w, Ilb-wll = radius of B. Also, using the remark

following le~ma ii, w is between a and b. Assume B is

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i0 SULLIVAN

internally tangent to U at z. Then, using lemma ii, b is

between ~ and z so that IIbll + llb - wll = i. We now show

by contradiction that b must lie between p and w. If p is

between b and w or p = b then, since p = g, I Ib-wll

IIb-gll + I]g-wll : ] lb -g l l + i/2(1-t 0) l i d - e l l - t O �9 Now

llg-wll > t o by the choice of t o . But then IIbII+llb-wll =

IIbll + IIb-gll + [Ig-wll m ]lgll + Ilg-w]l > (1-t O) +

t 0 = 1 a contradiction. Thus b must lie between p and w.

Therefore z must lie between d and e and so U is not

strictly convex at z. This is a contradiction to B

internally tangent to U at z.

It is an open question whether our main theorem

generalizes to higher dimensions. The author's conjecture

is that strictly convex must be replaced by the property

that the unit ball not "contain" a hyperplane in its

boundary. This property is equivalent to strictly convex

in the two dimensional case. This property also implies

that "concentric" is never an equivalence relation in the

one dimensional case which is easily established.

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SULLIVAN

REFERENCES

I. Clay, R. E. Contributions to Mereology, PhD. thesis, University of Notre Dame, Notre Dame, Indiana, 1961.

2. Sobocinski, B. Studies in Le{niewski's Mereology, Year book for 1954-55 of Polish Society of Arts and Sciences Abroad, vol. 5 (1955), pp. 34-48 London.

3. Sullivan, T. The Geometry of Solids in Hilbert Space, To appear, Notre Dame Journal of Formal Logic.

. Tarski, A., Foundations of the Geometry of Solids, A. Tarski, Logic, Semantics, Metamathematics, Clarendon Press, Oxford (1956) pp.24-30.

5. Valentine, F. A., Convex sets~ McGraw Hill, 1964.

Theodore Sullivan Department of Mathematics University of South Carolina Columbia, S.C. 29208

(Eingegangen am: 2. Oktober 1972)

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