Rend. Sem. Mat. Univ. Poi. Torino Voi. 52, 4 (1994)

N.V. Zhivkov

PEANO CONTINUA GENERATING DENSELY

MULTIVALUED METRIC PROJECTIONS*

Abstract. In every strictly convexifiable Banach space X with dim X > 2 a Peano continuum K exists such that with respect to an arbitrary strictly convex norm the metric projection generateci by K is multivalued on a dense subset of X. Moreover, there exists a starshaped Peano continuum S in X with this property provided dimX>3.

1. Introduction

Suppose (X, || • ||) is a strictly convex Banach space. A non-empty set M C X generates a distance function d(x,M) = inf{||a; — z\\ : z e M} and a set-valued mapping P(x,M) — {y e M : \\x - y\\ = d(x,M)} called metric projection or nearest poinl: mapping. It is known that P(-,M) is single-valued at most points, i.e. on a dense subset of G$ type, provided that M is a compact [St]. Zamfirescu [Za2] showed that P(-,M) might be multivalued on a dense set, as well, and that this is the typical case in the Euclidean space Rn when n > 2: In the Hausdorff metric space of compacta /C(Rn) most compacta (in sense of Baire category) generate metric projections which are essentially multivalued on everywhere continuai sets, and according to the proof of Stechkin's result they are single-valued on everywhere continuai sets too. Recali that a set is everywhere continuai in X if its intersection with any non-empty open subset of X contains continuum many elements.

Recently, De Blasi and Myjak extended Zamfirescu's result to the case of separable strictly convex Bahach space X, dim X > 2, and showed various arialogous theorems for different Hausdorff complete metric spaces of non-empty sets, [BM1], [BM2] and [BKM] (jointly with Kenderov), namely

- B{X) the space of bounded and closed subsets of X,

- JC(X) the space of compacta,

- C(X) the space of continua, i.e. connected compacta,

- S(X) the space of starshaped continua.

*Research partially supported by the National Foundation for Scientific Research at the Bulgarian Ministry of Science and Education under contract No MM58/91.

336 N.V. Zhivkov

A subsequent extension of a result of that type concerning IC(X) for arbitrary strictly convex Banach space X is obtained in [Zh2].

We shall prove here that in every strictly convexifiable Banach space X with dimA' > 2 a locally connected continuum K exists such that with respect to any strictly convex norm deflned on X the metric projection generated by K is multivalued on an everywhere continuai subset of X. It is shown that the constructed Peano continuum is a non-rectifiable curve of Hausdorff dimension 1. Moreover, if dimX > 3 then there exists a starshaped locally connected continuum S of Hausdorff dimension 2 with the same property while in a two-dimensional strictly convex space there is no starshaped Peano continuum with a densely multivalued metric projection.

A paper of Gruber and Zamfìrescu [GZ] is our motivation to consider fractal dimensions of continua. A theorem in [GZ] states that most elements of 5(Rn) have Hausdorff dimension 1.

It should be noted that according to [Ma] and [Bi] the Peano continua form a first Baire category set in C(X), and according to [Zal] the same is true in S(X), thus sets like K and S are not typical.

The paper consists of four sections including the present one. At the end of section 2 a general result about densely multivalued metric projections is established. In section 3 the basic planar construction which modifies an example from [Zhl] is given. Section 4 contains the main result.

2. Preliminaries

Let (X, || ||) be a strictly convex Banach space of dimension dimÀ' > 2. For x e X and e > 0, the open (respectively closed) ball with center x and radius e is denoted by B(x,s) (resp. B[x,e]). A line segment with end-points x,y e X ìs denoted by [x,y] and (x, y) means [x, 2/]\{tf, y}. The symbols l(x, e) and L(x, ei, e2) will stand for the line {.x + te : t E R}, ||e|| > 0, and the piane {x + tei + se2 : t,s e R}, ||ei||, ||e2|| > 0 respectively. For a subset M of X we denote by intM, bdM, coM and diamM its interior, boundary, convex hull and diameter respectively. If M is starshaped then kerM is its kernel. The Hausdorff distance between bounded and non-empty sets is designated by H.

Connected compacta are called continua and metrizable locally connected continua are referred to as Peano continua [Ku]. As a consequence of the classical theorem of Hahn-Mazurkiewicz-Sierpinski [Ku] a continuum is a Peano continuum if and only if for every e > 0 it is partitioned in a finite union of subcontinuua with smaller than e diameters, and if and only if it is a continuous image of the interval / = [0,1]. Hence the Peano continua from C(X) might be viewed as continuous curves taking values in X.

For s > 0, 6 > 0 and M C X, let

oo

HSS{M) = inf(^(dirnìUiY :\JUiDM,diamf/; < 6}.

Peano continua generating densely multivalued metric projections 337

To get the Hausdorff s-dimensional outer measure of M let 6 —> 0:

HS(M) = limWJ(M) = .supWJ(Af).

There is a unique value, dimM, called Hausdorff dimension of M (c.f. [Fa]), such that

HS(M) = oo if Ò < s < dim-M, Hs(M) = 0 if d i m M < s < o o

provided M is a subset of a finite dimensionai space. We denote the usuai dimension of a linear space and the Hausdorff dimension of a set by one and the same symbol.

DEFINITION 2.1. Let yo e M e X. À Une l = l{yo,e), \\e\\ - 1, is called a (two-sided) pseudo-tangent to M at yo if there exists a sequence (yn) C M, yn ^ yo far ali n, such that

(i)limyn = yo, (ii) \imd(yn,l)/\\yn - yo|| =0 , (Hi) the sequence ((yn — yo)/\\yn — i/o II) nas tw° cluster points e and —e.

It has to be noted that the notion of a pseudo-tangent is invariant under equivalent renormings of X.

For a set M G X, denote by A/^ (resp. A/^) the family of ali continuous images of J in M which are not points (resp. which are not line segments).

DEFINITION 2.2. Let N be a proper subset of a compact M. N is said to be a M°-separator (resp. A/"1 -separator) of M provided that N meets each element of Af^ (resp. Nix).

In order to prove a general result about the sets of multivaluedness of metric projections we need two technical lemmas:

LEMMA 2.3. ([Zh2]) Let x0,yo.6 X, \\xo - yo\\ = -d > 0 and l = l(yo,e) with ||e|| = 1 be a line such that {yo} = l fli3[a;o,d], and (yn) be a sequence satisfying

(i) \imyn = yo,

(ii)\imd(yn,l)/\\yn-yo\\=0.

Then there exists a sequence (xn) such that limxn = Xo and ||2/o — xn\\ =

ÌWn ~~ xn\\-

LEMMA 2.4. Let x0ìyo e X, dimX > 3, \\xo — yo\\ = d> 0 and l0 = l{yo,eo) with ||eo|| = 1 and l = l{yo,e) with \\e\\ = 1 be two lines such that eo and e are linearly independent and both lo and l are supporting the ball B[xo, d]. Let also (yn) be a sequence satisfying conditions (i) and (ii) front the above lemma. Then there exists a sequence (xn) such that lim.Tn = XQ and d(xn, lo) = \\xn — yn\\.

338 N.VZhivkov

Proof: Denote en = (yn - yo)/\\yn — 2/011- According to (ii) the sequence (en) has at most two cluster points e and -e . Without any change of indexes we consider the case lim(?/o + en) = yo + e. The other case is treated analogously.

Denote Ln(y0) '= L(y0ìe0,en), Ln(x0) = L(rE0,eo,en) and let (un) and (vn) be sequences such that {un} = P(yo,Ln(xo)) and {vn} = P(yn,Ln(xo)). Then {2/0} =.P(un,Ln(y0)), {yn} = P(vn,Ln(y0)) and \\y0 - yn\\ = \\un - vn\\. Obviously, II2/0 - un\\ < \\yn - un\\ and \\yn - vn\\ < \\y0 - vn\\. Hence d(unìlQ) < \\un - yn\\ and ||vn — yn\\ < d(vn,lo). Now applying the intermediate value theorem to the function <l>n(x) — d(x, lo) — \\x — yn\\ defined on [un, vn] we obtain a point xn e (un, vn) such that d(xn,k) = \\xn -2/nli­

ft has to be proved that \ìmxn = xo. Since (?/n) is a bounded sequence and un — #o + tn^o + snen for some tn,sn G R then (tn) and (sn) are bounded too. Let ito = #o + toeo + soe be a cluster point of (un). If wo 7̂ -̂ o then \\yo — xo\\ < \\yo — UQ\\ since {xo} = P(yo,L(xo,eo,e)). There is 6 > 0 so that \\yo — xo|| < II2/0 ~ wll whenever w G B(tio,<5). However, the last inequality contradicts the choice of un since for large n Il2/o -Eoli < Il?/o - Unii = d(y0ìLn(xo)). Therefore x0 = limwn = lim.Tn = l im^.

PROPOSITION 2.5. Suppose M is a compact which is nowhere dense in X and

(a) every point y G M lies on a pseudo-tangent to M at y,

(b) if y G (u, v) C M then there is a pseudo-tangent to M at y which does not contain u and v,

(e) there is a M1-separator N of M,

(d) the set Q = {x G X : P(x, M) fi N ^ 0 } is nowhere dense in X.

Then the metric projection P(-,M) is multivalued on a dense subset of X.

Proof: Let xoGX and e > 0 be given. Since M U Q is nowhere dense, we may assume with no loss of generality that B(xo, e) n (MUQ) = 0. Assume now, that P(-, M) is single-valued on B(xo,e). According to a result in [Si] the metric projection restricted on B(xo, e) is a continuous mapping.

We shall prove that ali points of B(xo,s) are mapped on a line lo passing through ?/o where {yo} = P(xo,M). If not so, there are Xi G B(XQÌS) with {yi} = P(XÌ,M)

for i = 1,2,3 such that co{yi,2/2,2/3} is a non-degenerated triangle. Having in mind the definition of A/"1-separator we conclude that the relative interior of co{.Ti,.T2,X3} is non-empty and for a point x from it the image of [x\,x] U [x,X2] via P(-,M) is contained in the line £(2/1,2/2 — 2/i)- By similar reasonings we see that x belongs also to l(y2,2/3 — 2/2) and / (2/3,2/1 — 2/3)» m u s proving that 2/1,2/2 and 1/3 are colinear.

Consider now the alternative:

(I) Either P{x,M) = {y0} for ali a; G B(x0,s),

(II) or there is x G -B(xo,£) with {?/} = P(x,M) such that for some u,v G M 2/€ (M,V).

Peano continua generatine densely multivalued m'etric projections 339

In case (I) let / be a pseudo-tangent to M at i/o. Obviously, / is a supporting line for the ball B(x0,d), d = d(xQ,M) > 0, i.e. I n B(x0ìd) - {yo}. Applying Lemma 2.3 we obtain a sequence (xn), lim#n = x0 such that \\y0 - xn\\ = \\yn - xn | |. But now, for large n Xn belongs to B(x0,e) and this violates the single-valuedness assumption.

The case (II) is treated analogously after applying Lemma 2.4. The proof is completed.

COROLLARY 2.6. Suppose M is a nowhere dense compact with pseudo-tangents at allpointsand there is a N°-separator N ofM such that the set {x G X : P(x, M)C\N ^ 0 } is nowhere dense too. Then the metric projection P(-, M) is multivalued on a dense subset of X.

The case of a totally disconnected compact M is considered in [Zh2].

3. Planar Construction

In the Euclidean piane (M2, | • |) a Cartesian coordinate system Oxy is used. The closed unit disk(ball) is denoted by B.

Let An = [Pnl, Pn2ì—ìPn2n] be a regular 2n-gon inscribed in B with vertices

Pni •= (cos(7r(» - l ) ^ " 1 ) ^ ^ ^ - 1V2"-1)) for i = l,2,.. . ,2n , n > 1. For qn = sin(7r/2n)/(l 4- sin(7r/2ri)) let Dni = (1 - qn)Pn% + qnB, i-e. Dni are the images of B via homothets with centers Pni and scaling factors equal to qn. Defìne

2"

Mn = ( j D n ^ X n - { D n i : i - l , 2 , . . . , 2 n } , n = l,2,.... t = i

The scalarsi qn are chosen such that any two adjoining disks frpm Mn have only one boundary point in common and thus Mn are continua.

Now, a Peano continuum K CR2 will be cpnstructed as a countable intersection of a nested family of continua Kn such that for each n the set Kn is a finite union of closed disks with equal radii ?v

Assign KG = B, /C0 = {B} and r0 = 1. We are going to define Kn and /Cn under the inductive assumption that Kj and K,j have already been defined, so that the disks of /Cj have radii r$ = 4142—97 for j = 1,2,..., n — 1. (See the figure visualizing the sets K0, Ku k2andK3.)

For D e /Cn_i let TD be the translation carrying O to the center of D. Set

#n - [jiTnir^iMn) : D G Kn^}

and

/Cn = {^ ( r^ - iE ) : D G !Cn^Ee Mn}-

The elements of /Cn are closed disks with radii equal to rn = q\q2---qn- Put finally

340 N.V.Zhivkov

Figure. The union of filled disks is K3.

The set K is a compact as an intersection of a nested sequence of compacta. It is easily seen that intK = 0 and

(3.1) lim H(Kn,K)=Q. n—»oo

With K we associate sets Vn, Nn for n > 1 and also V and N. 00

Vn = {TD(rn-iPni) : D 6 /Cn_i, i = 1,2,...,2"}, If = [j V" n = l

oo'

Nn = {x e Kn : 3DU D2 e £ n , Dx ^ D2ì {x} = Dx n Z>2}, AT = (J JVn. n=l

The properties of .K" are collected in the subsequent lemmas.

LEMMA 3.1. K is a Peano continuum.

Proof. It should be established at first that K is a connected set. Having in mind (3.1) it suffices to prove that Kn are continua. It is seen by induction that Kn is connected if and only if Nn-i C Kn because Mn is connected. Let x e Nn-i and m < n be the least integer for which there are disks D\,D2 e JCm with centers c\ and c2 such that D\ ¥" D2, {%} = Di f)D2 and both £>i and D2 are contained in a disk from K,m-\. Theri [ci,C2] is parallel to a side of Am. There is a vertice v of Am+i which is perpendicular to that side, hence there are two vertices v\ and v2ì v\ = -v2 both parallel to [c'i,C2]. Then the images of {^1,^2} via rmTDi, i = 1,2, meet the middle point x = (c\ + C2)/2.

Peano continua generating densely multivalued metric projections 341

In order to verify the locai connectedness of K we observe that for every n and every disk D of/Cn the set DnK is a continuum, thus K'is partitioned in a finite union of continua with diameters 2rn, but rn tends to 0 when n increases. The proof is completed.

It is seen from the proof of Lemma 3.1 that N'cV.c K.

LEMMA 3.2 N is a N°-separator of K.

Proof. It is sufficient to observe that for each n > 1 Nn separates Kn, i.e. the set Kn\Nn is not connected.

LEMMA 3.3. K is a non-rectifiable curve.

Proof. Let K be parameterized by a continuous mapping (p : I —>. R2, i.e. (p(I) = K. For a fixed integer n > 1, let T = {ti < ti < ... < tm} be a partition of / such that (p(T) = Vn+i and for each % = l,...,m — 1 <p(U) ^ ip(ti+i). Up to a subpartition of T we may assume also that any two adjoining points ti and t?;+1 are mapped by tp in one disk from Kn. This is so, because Nn separates Kn and JCn is a finite set. If (p(ti) and (p(U+i) belong to different disks for some i, then there is a finite set of reals between ti and ti+i such that the images of any two neighboring numbers are in one disk. Therefore,

m—1

i= l

where o~n = 2(1 + sin(7r/2n+1)) Ifltl ^Qi ls t n e s u m of the perimeters of ali 2n+1-gons inscribed in disks of JCn. However,

^ TT 2n sin(.7r/2n). lim (jn = 2 I I ——. \ '. \ n^oo n 1 1 i _j_ sin(7r/2n)

n=l \ i /

and the infinite product is not convergent since the n-th multiplier does not approach 1.

LEMMA 3.4. The Hausdorjf dimension of K is 1.

Proof Since K is not a point then dirnif > 1. On the other hand, making use of monotonicity of H^(K) with respect to 6, we write:

7is(K) = lim HL(K) < lim V{(diam£>)s : D e 7Cn}

n(rt. + l )

n — • o o "• n—>oo

n

lim T-^i— (2rn)s = 2S lim 172*^ = 2a lim T] di{s)bi(s)

n—*oo n—»oo ••••*•• n—>oo •*--••

where a*(s) = 2isin(7r/2i)/(H-sin(7r/2i))s and ò*(s) = s i n * - 1 ^ * ) . Since lim a»(s) = ir, and for s > 1 limò*(s) - 0, then ^ ( i f ) = 0 for 5 > 1. Therefore dimK = 1.

LEMMA 3.5. Every point yo G K lies on a pseudo-tangent.

342 N.V.Zhivkov

Proof. Let D E /Cn_i, D' E /Cn, D" E JCn+1 and y0 € Z>" C £>' C •£>. Let also w,wi,V2 € Vn n .D, V E D" and vi and i>2 be v's two neighboring vertices. It will be used later that v\ and V2 depend on n, i.e. v\ = ^i(n) and V2 . == 'U2(w).. Denote by n̂ = KV0ìen)y |en| = 1» the straight line through y0 which is parallel to' [vi,^]- A'n

elementary calculation shows that

\VÌ — 2/o| >rn-ism(7r/2n~1)-rn+iì i = 1,2

On the other hand for Iarge n [vi, V2] does not intersect D", i.e. rn_i(l—cos(7r/2n-1)) > 2rn+i, and d(vi,ln) < rn_i(l - cos(7r/2Tl~1)). Hence

( 3 2 ) rffeLW< l - c o s ^ " - 1 ) sin(7r/2".). . • K - 2/01 ~ sin(7r/2n-1) - gng„+i cos(7r/2n) - anqn+i

with on = 2 - 1 ( l + sin(7r/2n))_1. The expressions in (3.2) tend to 0. Let e be a cluster element of the sequence (en). Without any change of indexation

we assume that limen = e. Denote v\n = \en — e | and / = l(yo,e). Let y ^ yo and un £ P(yJn)- The following estimation for distances holds:

d{y, l) <\y-Un\+ d(un, l) <\y ~ un\ + \un - yo\.\en - e| <

(3.3) (1 + Vn)d{y, ln) + Vn\y~ 2/o|-

Defìne a sequence (?/„):

{ _ vi (m) for n = 2m - 1

Vn ~~ ^ U2(m) for n = 2m.

then by (3.3) d{ynJ) < (1+^)^4+^,

l^n — 2/o| fon" 2/01

where m is the integer part of (n + l)/2. Having in mind (3.2) we see that the expressions tend to 0 when n increases. It is

a routine matter to verify that (yn — yo)/\yn - 2/oI has two cluster points. Therefore / is a pseudo-tangent.

LEMMA 3.6. Every line through every element yo of N is a pseudo-tangent to M.

Proof. Let n > 1 be an integer and {yo} = £>i n D2 for Di, D2 E Kn. Denote by lo the line which separates Di and D2. Let lni be the lines through yo which make elementary angles equal to m/2n, i = 1,2, ...,2W_1, with lo. These lines intersect V at infinitely many points which approach yo from both sides. Obviously, lni are pseudo-tangents. Thus there is a dense set of pseudo-tangents. In order to prove that any line through 7/0 is a pseudo-tangent we make use of (3.3).

Peano contìnua generating densely multivalued metric projections 343

4. General Case

THEOREM 4.1 Suppose X is a strictly convexifiable Banach space.

(I) IfdimX > 2 then there exists a Peano continuum K of Hausdorffdimension I in X such that for any strictly convex norm in X the metric projection generated by K is multivalued on a dense set which is everywhere continuai in X,

(II) IfdimX > 3 then there exists a starshaped Peano continuum S of Hausdorff dimension 2 in X such that for any strictly convex norm in X the metric projection generated by S is multivalued on an everywhere continuai subset of X.

Proof. Let K b e a two-dimensional subspace of X and / : M2 —> F b e a linear isomorphism. Denote by K the image of the constructed in the preceding section Peano continuum Ky i.e. K = f(K). Denote also N = f(N), V = f{V) etc. Suppose || • || is a strictly convex norm in X. There are positive constants c\ and c2 such that for any x,ye R\ d\x - y\ < \\f(x) - f{y)\\ < c2\x - y\.

Ali the properties of K are shared by K. In order to verify that dim K = 1 we make use of the next

LEMMA 4.2 ([Fa] Lemma 1.8) Let f : Mi ——> M2 be a surjective mapping such that \\f(x) - f(y)\\ < c\x~ y\ x,y € Mi far a Constant e. ThenHs(M2) < csH8(Mi).

Trivial checks establish the other properties of K analogous to those of K listed in lemmas 3.1-3.6. From now on the parts (I) and (II) are considered separately.

Part (I): In order to apply Corollary 2.6, we have to show only that intQ. = 0, where Q = {x e X : P(x,K) fi N ^ 0} . For this purpose let x e Q, y e P(x,K) fi N and C(x) = Y n B[x, d(x,K)]. Since any line in Y through y is a pseudo-tangent, then C(x) has an empty relative interior in Y. Thus

d(x, K) = d(x, Y) whenever xeQ.

If Q is dense in some open and non-empty set U then the above equality holds for ali points in U, but this is impossible since K is a nowhere dense compact and || • || is strictly convex. ;

Denote by W the set of points of multivaluedness of P(-, K). According to Corollary 2.6 W is dense in X. In order to show that W is everywhere continuai in X we employ the following

LEMMA 4.3 ([Zh2]) Let Mi and M2 be two disjoint and non-void compacta and x € X be a point such that d(x,M\) = d(x,M2) Then arbitrary neighborhood of x contains continually many elements with the some property.

Suppose x e W\Q and yi,y2 e P(x,K), \\yi - y2\\ = d > 0. There is a sufficiently large integer n so that y\ and y2 belong to two different sets D\ and D2 from

344 N.V. Zhivkov

)Cn. Since G = bdDi n Nn is a finite set (in fact it contains no more than three points) and G n P(x,K) = 0 then there is 6 > 0 and an open neighborhood E/ of x such that M = K\(JveGB(v,Ó) does not contain ^ , i = 1,2, and both P(-,I?) and P{-,M) coincide on U. It remains to apply Lemma 4.3 in order to complete the proof of (I).

Part (II): Since dimX > 2 there is e e X\Y. Denote by Z the linear span of {e} U Y and consider the set

S={t(e + y):yeKìte[-lìl]}.

It is a starshaped Peano continuum of Hausdorff dimension 2 with kerS = 6, where 6 is the origin of X. The set N = {t(e + y) : y e N,t G [-1,1]} is a A/"1-separator of S. In applying Proposition 2.5 we need verify condition (d) only, i.e. Q = {x G X : P(x, S)C\N =̂ 0 } is nowhere dense in X. If not so, then there is an open set U e X such that UnS = <2> and Q is dense in U. Denote E = ({6}U(e + K)U(-e + K)) and consider the alternative:

(i) Either P(.T, 5) C E whenever x G U,

(ii) or there is an open and non-empty set U\ <Z U such that P(x, S) C S\E whenever x G U\.

The case (i) has been treated already. In the Iatter case observe that for every y G N\E any line through y which is contained in Z is a pseudo-tangent to S at j / , hence

(4.1) d(x,S) = d(xtZ) whenever x G U\.

However, S is nowhere dense in Z and (4.1) does not hold for ali sufficiently small and parai lei to Z shifts of a:.

For the remaining part of the proof our reasonings are very similar to those concerning part (I) when proving that the set of points of multivaluedness of the metric projection is everywhere continuai. We need only distinguish between two separate cases again: P(-, S) maps an open set U on E, and P(«, S) maps an open set U on S\E. The proof is completed.

The next result shows that the minimal dimension at which (II) holds is 3.

PROPOSITION 4.4. Suppose S is a starshaped Peano continuum in a two-dimensional strictly convex space X. ìf the metric projection P(-,S) is multivalued on a dense subset of some open set U C X, then S has non-empty interior.

Proof. Let P(-,S) be multivalued at some XQ e U. Since the norm is strictly convex, then ker S fi P(XQ, S) = 0. For y0 G P(xo,S) denote xt = (1 - t)xo + ty0 and choose t > 0 sufficiently small so that xt G U. It follows from a well-known lemma (c.f. [St]) that {yo} = P(xt, S). Let (xn) he a converging to xt sequence such that the metric projection is multivalued at èach point xn. Take ZQ e ker S. For every integer n there is

Peano continua generatine densely multivalued metric projections 345

yn e P(xn,S) such that yo,yn and z0 are convexly independent. The continuity of P at xt implies \imyn = y0.

Let now e = 112/o - ^o||/2 > 0. Since S is locally connected at yo, then for a suffìcierttly large number n a continuous patii in SnB(yo,e) connect yo with yn. Therefore, co{zo,yo,yn}\B(yo,e) is a subset of S, and obviously it has non-empty interior.

Thus the metric projection P(- ,5) is single-vaiued on an open set.

REMARKS. It has to be mentioned that theorem 4.1 provides new information not only for non-separable Banach spaces since most continua in C(X) and S(X) are not Peano continua. According to a result of Mazurkiewicz [Ma], later extended by Bing [Bi], the (hereditarily) indecomposable continua form a dense G$ subset of C(X). A continuum is indecomposable if it cannot be represented as union of two proper subcontinua. It is seen that an indecomposable continuum is not locally connected at any of its points. This is no longer true for continua in S(X) since the starshaped sets are locally connected at the points of their kernels. However, an analogous result of Zamfìrescu [Zal] stili holds and it shows that the set of Peano continua in S(X) is of first Baire category (although formulated for 5 (R n ) the proof of Zamfìrescu combined with [BKM] works in arbitrary Banach space). Hence the existence of sets like K and S even in R2 and R3 respectively is not guaranteed by any of before quoted theorems.

REFERENCES

[Bi] BiNG R.H., Concerning hereditarily indecomposable continua. Pacific J. Math. 1, 1 (1951), 43-52.

[BKM] D E BLASI F.S., KENDEROV P. AND MYJAK J., On the metric projection onto compact starshaped sets. Monatsh. Math. (to appear).

[BMl] D E BLASI F.S., MYJAK J., Ambiguous loci of the nearest point mapping in Banach spaces. Arch. Math. (to appear).

[BM2] D E BLASI F.S., MYJAK J., On compact connected sets in Banach spaces. Proc. Amer. Math. Soc. (to appear).

[GZ] GRUBER P., ZAMFÌRESCU T., Generic properties of compact starshaped sets. Proc. Amer. Math. Soc. 108 (1990), 207-214.

[Fa] FALCONER K.J., The Geometry of Fractal Sets. Cambridge Tracts of Math. 85, Cambridge University Press 1985.

[Ku] KURATOWSKI K., Topology voi. II (1968), PWN Academic Press, New York and London. [Ma] MAZURKIEWICZ S., Sur les continus absolument indecomposable. Fund. Math. 16 (1930),

151-159. [Si] SINGER I., Some remarks on approximative compactness. Rev. Roum. Math. Pures Appi. 9

(1964), 167-177. [St] STECHKIN S.B., Approximative properties of sub sets of Banach spaces. Rev. Roum. Pures

Appi. 8 (1963), 5-8. [Zal] ZAMFÌRESCU T., Typical starshaped sets. Aequationes Math. 36 (1988), 188-200. [Za2] ZAMFÌRESCU T., The nearest point mapping is single-valued nearly everywhere. Arch. Math.

54 (1990), 563-566. [Zhl] ZHIVKOV N.V., Examples of piane compacta with dense ambiguous loci. Compt. rend. TAcad.

346 N. V Zhivkov

bulg. Sci., 46 (1993), 27-30. [Zh2] ZHIVKOV N.V., Compacta with dense ambiguous loci ofmetric projections and antiprojections.

Proc. Amer. Math. Soc. (to appear).;

N.V. ZHIVKOV Institute of Mathematics, Bulgarian Academy of Sciences Sofia 1113, Bulgaria.

Lavoro pervenuto in redazione il 28.6.1993.