ON T H E T O P O L O G Y OF T H E E F F E C T I V E S U B S E T S OF LEVEL SETS

L. E. Baz i l ev l ch UDC 515.12

In studying the problem of constructing selections of the distance function to subsets of a Euclidean space the concept of the effective subset of a level set arises In this paper we establish the connectivity' of effective subsets, which is important for applications.

The topology of the level sets of the distance function to subsets of a Euclidean space has been studied by many authors [3-5]. We note in particular the article of S. Ferry [4], which contains an exhaustive study of the problem of homeomorphisms of level sets with manifolds. In the author's papers [1, 2], which are devoted to the construction of selections of the distance function to compact subsets in N n, an important role is played by a special subset of a level set - - the effective subset F(A, r) (for the definition cf. below).

In the present paper we establish the connectivity of the effective subset, which is essentiM in applica- tions.

1. Def in i t ions a n d l e m m a s . Let [a, b] be a closed interval. We shall use the notation [a, b[ := [a,b] \ {b}, ]a,b] := [a,b] \ {a}, and ]a, b[=]a, b] \ {b}. The symbols (a,b) and [a,b) denote respectively the line passing through the points a and b and the ray with origin at the point a passing through the point b. Let K ( a , R ) : = {x E N~ld(x,a) < R}, R > O, be an open ball and T(a,L,c~) the ~-cone with vertex a over the set L---the set of points t E 1~ ~ such that the angle / tap is not larger than c 2 for some point p E L.

We define the complete projection of the point x E ]R n \ A to be the set P(x) := {y E A! d(x, y) = d(x,A)} and call a point p(x) C P(x) a projection of the point x. The interval [x,p(x)[ is called the projection interval with base at the point x and end at the point p(x). Given two projection intervals, either they do not intersect, or they have a only a common endpoint, or one is contained in the other. A projection interval that is not contained in any other projection interval is said to be maximal. We denote

by K~ (~) the ball with diameter ]x,p(x)[. The symbols BdxA , CIxA, and In txA denote respectively the boundary, the closure, and the interior of the set A in the topological space X. If X = R '~, the symbol X will be omitted from this notation.

L e m m a 1. Let x E R" \ A and p(x) e P(x). Then for every y E BdK~ p(~) \ {x} the distance function to A is strictly increasing as a point moves from y to x on the interval [y, x].

PROOF. Assume there exist points tl and t2 E [y,x] such that d( t l ,x ) < d(t2,x) and d(t l ,A) < d(t2,A). Let M := {z E Rnld( t l , z ) < d(t2,z)}. It is obvious that P(t l ) C M. Therefore d( t l ,A) > d ( t l , M \ K ( x , d ( x , A ) ) > ld iam ( B d M fq/(~(~)) > d(t2,p(x)) > d(t2,d). Contradiction. The lemma is now proved.

Coro l la ry . Let pl(z) and p2(z) be two projections of the point z such that the size ~ of the angle gp~(z)zp2(z) is less than 7r. Then for each r e [Rsin if, R[, where R = d(z ,A), the intersection of the r-level E (A , r ) with the triangle ~pl(z)zp2(z) is a simple arc with endpoints on the intervals ]z,p1(z)[ and

PaooF. It follows from Lemma 1 that for every y E ~pl(z),p2(z)] the distance function to the compact set A is strictly increasing on the interval [y,z]. Since max{d(y ,A) ly E ~pl(z),p2(z)]} < Rsin ~, the corollary is now proved.

We define the v-level of the distance function to the set A (r > 0) to be the set E(A , r ) := {x E R~ld(x ,A) = r}. We also introduce the notation E - ( A , r ) : = {x E Nnl0 < d(x ,A) < r}, E + ( A , r ) : = {x E R" I d(x,A) > r}. We now fix some connected component E+~(A,r ) of the set E+(A,r) . We define the

Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 5-9. Original article submitted October 3, 1988.

0090-4104/93/6405-1117 $12.50 @ 1993 Plenum Publishing Corporation 1117

effective subset of the set E+~(A, r) to be the set F~(A, r) = F(A, r) all boundary points of E+~(A, r) that are reachable from within +~ E (A,r), and we set E ~ ( A , r ) : = BdE+~(A,r).

L e m m a 2. Let x E IRa\ A, p(x) E P(x), and y E]x,p(x)[ME~(A,r) forO < r < d(x,A)). Then +~ x E E (A,r) .

P r o o f . It follows from Lemma 1 that there exists a neighborhood U of the point y such that all points in the set U fl E+(A, r) can be joined to the point x by an interval contained in the set E+(A, r). Consequently U meets only one connected component of the set E+(A, r). But since y E E~(A, r), this component is E+~(A, r). The lemma is now proved.

L e m m a 3. The set of points in E~(A, r) that are not endpoints of maximal projection intervals is everywhere dense in the set E~(A, r).

Pl too r . Let x E E~(A,r) , p(x) E P(x), and let y E K(x, ~) M E+~(A,r) be arbitrary (r > 0). Set z = [y,p(y)[ME(A,r). It is obvious that z E E~(A,r) and d(x,z) < e. Since e is arbitrary, the lemma is proved.

Coro l l a ry . The set of points in E~(A,r) that are not endpoints of maximal projection intervals is contained in the set F(A,r); consequently F(A,r) is everywhere dense in set E~(A,r).

L e m m a 4. Let c E N n \ A, p(c) E P(c), and let R E]O,d(c,A)] be a certain number. Then there exists r > 0 sufficiently small that for any r E]0,r there exists a(~) E ]0, ~[ such that for any y E K(c,e) N T(c, {p(c)},a(e)) and any pl(y),p2(y) E P(y) the angle Ap,(y)yp2(y) is less than 2arcsin n RTe

T: where a(c) --+ -ff as e --~ O.

PROOF. We denote by B( r the set of points x E R '~ \ A for which the angle /pl(x)xp2(x) is smaller than the number r for any pl(x),p2(x) E P(x).

We set ~ = ~ - a r c s in n and pass a ray I through the point c making angle a, a E ] 0, ~ -29~ [ with the

ray [c,p(x)). Let z = IMK~ (c). For any y E ]c,z[ we have P(y) E T(y, {z},fl) \ K(c,d(c,d)), where/3 is the size of the angle Zzyp(c). Let z0 be the point of intersection of I with the line passing through the point p(c) and making angle c 2 with the segment [z,p(c)] inside the triangle Aczp(c). Then B(2arcs in XB47+~) D [z0,c[. It remains only to require that the condition d(zo, c) > r hold, i.e., ~ + r sin a tan c~ < r cos a. As e ~ 0 and q0 --~ 0, therefore, the last inequality will hold starting from some r > 0. The lemma is now proved.

T h e o r e m 1. The effective subset F(A,R) is arcwise connected.

PROOF. Assume that F(A, R) is not arcwise connected. Take an arbitrary x E F(A, R) that is not the endpoint of a maximal projection interval (corollary to Lemma 3). As shown in [4], there exists a neighborhood V of the point x such that the set V M F(A, R) is homeomorphic to an (n - 1)-dimensional disk and consequently arcwise connected. Denote by N1 the arc-component of the set F(A, R) containing x. It is obvious that V Cl F(A,R) C N1. Set N2 := F(A,R) \ N1. It is easy to see that d(N~,N2) = O.

Fix a number r > 0 that is very small in comparison with R and set S := {z E E - ( A , R + r E+~(A, R)I d(z, Na) = d(z, N2)}. The set S is closed in E-(A, R+e)ME+~(A, R) and consists of points that are endpoints of maximal projection intervals. Moreover SME-(A, R+6) 7s 0 for any 5 E ]0, e[. It is easy to see that d(z,E(A,R)) = d(z ,E~(A,R)) = d(z ,F(A,R)) = d ( z , A ) - R , and therefore d(z, N1) = d(Z, N2) = d ( z , g ) - R for z E S. Consequently the subsets Pi(z) C P(z)and Pi(z) := {p(z) E P(z)l [z,p(z)[ Mgi 7 ~ 0}, i = 1,2, are nonempty. It is obvious that Xi(z) E g i , where Xi(z) := {[z,p(z)[NE(A,R)ip(x ) E P(x)}, i = 1,2. As can be seen from the corollary to Lemma 1, for arbitrary x~ E Xa(z) and x2 E X2(z) the

R R angle /x lzx2 is larger than 2 arcsin d-U-~,A ~ > 2 arcsin ~4-~' i.e., because r is small this angle does not differ

markedly from a straight angle. We shall show that for any z E S there exists a neighborhood U ~ z such that U fl S is homeomorphic

to an ( n - 1)-dimensional disk.

It is easy to see that diamPi(z) _< 2d(z,A)sin~o = 4d(z,A) n~(R+~)~ - < 4 R ~ n + e := k (q0 =

2arcsin R---~)" Let p*(z) E Pi(z) be points such that Pi(z) C Cl(p*(z),k). Without loss of generality we shall assume that the point z coincides with the origin and the ray [z,p~(z)) points in the direction of the positive Oxn-axis. As follows from the fact that a distance function is continuous, for an arbitrary

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> 0 there exists ~ < ~ such tha t P(y) C E-(P(z),e) C K(p~(z),k + e) U K(p~(z),k + r an arbitrary y E K(z,6). We set Ai := ANK(p*(z) ,k+e) , i = 1,2. We have P(y) C A1 UA2 for y E K(z , 3).

We define U := K(z, 3) n E-((z,p;(z)),3/v~). We denote by l v the line pass ing th rough the point y E U parallel to (z,p~(z)) and we set ]Yl,Y2[ := Iv VI K(z,3), where d(yl,p~(z)) < d(yz,p~(z)).

We now make the following remark. In order for the dis tance funct ion to an arbi t rary set M to be strictly increasing (resp. decreasing) on an interval [ya, y2] it suffices tha t the dis tance funct ion to each point m E M have the same p roper ty on the interval. This last condi t ion holds when all the points x,,, E (Yl, Y2) such tha t d(Iv,m ) = d(xm, m), m E M, belong ei ther to the ray [yl,y2)\]yl,y2[ or to (yl,y2]\]yl,y2[.

Subs t i tu t ing the sets A1 and A2 in place of M , and taking a sufficiently small e (so tha t the relations 4 R ~ 2~ + k = 2~ + R+, < R holds), we conclude tha t for any y E U the dis tance funct ion to A1 is strictly

increasing and the dis tance funct ion to A2 is str ict ly decreasing as a point moves f rom yl to Y2 on the interval [Yl, Y2].

We now define W := {y E U[d(y, A1)= d(y, A2)}. It is obvious tha t d(y, A1)= d(y, A2)= d(y,A) for y E W, i.e., P(y)VIAi # 0 for i = 1, 2. As follows f rom L e m m a 1, for any y E U we have d(yl, At) < d(y~, A2) and d(y2,A2) < d(y2,A1). Therefore there exists a unique point wv E]Yl,Y2[ such tha t d(wv,A1) = d(wy, A2).

We now identify R '~-~ wi th the subspace Q := { ( x l , . . . , xn) E R'* I x~ = 0} and let 7rn : i$~ ~ -+ R ~-~ be the or thogonal project ion. We use the no ta t ion K -- ~ru(U). We have a m a p p i n g ~a : / ( --~ W, T(y) = wy which, as one can easily see, is a homeomorph i sm. Consequent ly the point z has a ne ighborhood U C K(z, 5) (5 < ~ being some number ) , in which the set U M S = W is homeomorph ic to an (n - 1)-dimensional open disk. We shall now show even more, tha t for all z E S M E-(A, ~) one Call set 3 = 2, i.e., the intersection

S V1K(z, ~) is an (n - 1)-dimensional disk. Let I := l~. We denote by o. a maximal posit ive number such tha t in the cylinder D wi th base K(z, ~)MQ

and axis l the opera t ion of projec t ion along l is one-to-one on the set S ~ --- S M D M K(z, R). Since each point of the set S ~ has a ne ighborhood in which this set is homeomorph ic to an (n - 1)-dimensional disk, we conclude tha t the whole set S ~ is homeomorph ic to an (n - 1)-dimensional disk.

From the maximal i ty of the n u m b e r a it follows tha t there exists a point We now assume tha t ~ < ~. y E Bd (K(z, e) N Q) such tha t on the line ly passing th rough y parallel to l there is e i ther no point or at most one point of the set S.

We now pass a plane II t h rough the line 1 and the point y. It is obvious tha t the set H n D 7~ S is a simple arc 7, and 7 can be represented as the graph of some cont inuous funct ion in the plane II, taking as x-axis the line II V1 Q and as y-axis the line l, the uni t of length being induced f rom I~ '~. T h e n 7 = 7(t), t E] - a, a[, and the point y has coordinates (a, 0) in the plane H.

We consider the set F := Jim, 7(t). Two cases are possible: ei ther F is a closed interval, or F is a point.

Let us consider the first case: F = [a, b], a # b. It is obvious tha t the dis tance funct ion to the compact set A is cons tant on [a, b], d(c,A) E [R,R +r for c E [a, hi, and therefore, as follows f rom L e m m a t , all the project ion intervals wi th endpoin ts at the points c E ]a, b[ are perpendicu la r to this closed interval, i.e., they lie on the (n - 1)-dimensional hyperp lane Qr passing th rough the point c and parallel to Q. Since c E C1 S, the point c has two project ions pl(c) and p2(c) such tha t Lpl(c)cp2(c) > 2arcs in /t - d ~-KT~,A~" But by L e m m a 4 the project ion intervals ]pi(c), c], i = 1, 2, make an angle a ' E [a(r ~- - a(r wi th the interval Qc gl II gl D. We denote by D' the par t of the cylinder D tha t is contained between the balls K(p(z), R), p(z) E P~(z), and the balls K(p(z),R), p(z) E P2(z). As follows from L e m m a 4, we have S' C D ' \ T(c, {p(c)},a(r But since a(r ~ ~ as ~ ~ 0, for sufficiently small ~ the projec t ion of the set D' \ T(c, {p(c)}, a(r along the line l does not fill up the entire ball K(z, a) M Q; consequently we have ob ta ined a contradic t ion to the choice of the n u m b e r ~r.

Thus the set F is a single point 7(o'). By the choice of a we have 7(c,) ~ S; consequent ly 7(a) E E(A, R). We now fix the points x~ E Xj(z), j = 1, 2, and show tha t they can be joined to the point 7(~) by an

arc in the set F(A, R). Fix i = 1, 2. The funct ion g(t) = d(7(t),A),t E [0,3], assumes values on the interval ]R,R + ~[, and g(t) --* R as

oo t ~ 3. We now choose a sequence of posit ive numbers { ,~},=l t ending monotonica l ly to zero with ~ < d(z, A) - R, and let {tn } ~=~ be an increasing sequence such tha t t l = 0 and t,~ = max{t E [0,511 d(7(t) , A) =

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R+~n} for n > 2. Since lim 7(t) = 7(6) is a single point, it follows that diam {7,, = 7(t), t E [tn, tn+l]} ~ 0 t--,6

aS r t - - + OO.

For an arbitrary n we choose points x~' E Xi(7(t , ) ) . We fix some n E N and use the notation = e [*.,*.+1]}. We have ai X;' < d i a m T , + 2e,~. Let Y = U{['y(t),T]IT E convXi(t)}, t E [tn,tn+a]. We have Y C T(7(t),{*},~), where ~ <

R 2arccos k--4u and * is some point. We set )(i(t) = Y M E(A,R). It is obvious that Xi(t) D Xi(t). We set ~'~ = U{Xi(7(t))lt E [tn,t,+l]}. It is easy to see that the compact set .~_ i s connected and x~ and x~ +1 belong to Z~ C X['. We have diamX'~ < diam7~ + Cn. All points of X'2(t ) are noncritical, and therefore there exists a domain U(t), Xi(t) C U(t) C F(A, R) M E(Xi(t), r homeomorphic to an (n - 1)-dimensional disk. From the covering {U(t)lt E [t,,tn+]]} of the continuum dr{' we choose a finite

k subcovering U1, U2,..., Uk, and in the set U Uj we join the points xp and x~ '+1 by an arc a~. It is easy to

j = l

see that diamap < diam-y., + 5r Then the set {7(6)} U {apl n E N} is the required arc joining the point z* to the point 7(6) in the set F(A, R). Consequently the points z~ and z~ can be joined by an arc in the set F(A, R). We have now obtained the required contradiction.

Thus for an arbitrary point z E SME-(A ,R+ 2) the intersection SMK(z, 2) is an ( n - 1)-dimensional disk for which there exists a line l such that the projection of the disk along the line 1 onto the hyperspace Q orthogonal to it defines a homeomorphism.

Z oo We now construct the sequence { -}n=l, zi E S for n E N as follows. Let zl E S [ '1E- (A,R + ~) be an arbitrary point. If ml := min {d(y,A)ly E S Cl S d K ( z l , ~)} < d(zl,A), we choose the point z2 in the set {y E S FI B d K ( z l , 3)ld(y,A) = rnl}, while if rnl >_ d(za,S), we let z2 E S M E - ( A , kl), where hi = min{d(y, A)I Y E S FI CI K (z, }) }. It is obvious that K (zl, }) M K (z2, }) = O. Assume that the point Zn has already been constructed. If rn, := min {d(y,A)ly E S M K(zn, ~) } < d(zn,A), we take z,~+l from the set {y E SMBd(z , ,~)Id(y ,A ) = m , } , and if m,~ k d(zn,S), then welet zn+a E SME-(A,k , ) , where kn := min {d(y,A)l y E S M C1K(z, }) }. It is obvious that K(z,+l, 6) M K(zi, 6) = ~ for all i, 1 < i < n.

Thus we have constructed a countable set of pairwise disjoint balls K(zj, 6) of identical radius. But this is impossible, since all the balls K(zj, ~) lie in the bounded set E-CA , R + e). The theorem is now proved.

L i t e r a t u r e C i t ed

1. L. E. Bazilevich, "On selections of the distance function to a compact set," Vestn. L 'roy Univ., Ser. Mekh.-Mat., No. 24, 54-59 (1985).

2. L. E. Bazilevich, "On selections of the distance function to a compact set in Euclidean space," in: Proc. Baku Internat. Conf. Topology [in Russian], Kommunist, Baku (1987), Pt. 2, p. 27.

3. M. Brown, "Sets of constant distance from a planar set," Mich. Math. J., 19, No. 1,321-323 (1972). 4. S. Ferry, "When e-boundaries are manifolds," Fund. Math., 30, No. 3, 199-210 (1976). 5. R. Gariepy and D. W. Pepe, "On a level set of a distance function in a Minkowski space," Proc. Amer.

Math. Soc., 31, No. 1,255-259 (1972).

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