Gated sets in metric spaces | SpringerLink

Aequationes Mathematicae 34 (1987) 1 1 2 - 1 2 0 University of Waterloo

0001-9054/87/0011124)951.50 + 0.20/0 © 1987 BirkhS.user Verlag, Basel

G a t e d s e t s in m e t r i c s p a c e s

ANDREAS W. M. DRESS a n d RUDOLF SCHARLAU

Dedicated to Professor Otto Haupt with best wishes on his lOOth birthday Abstract. The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.

1. Introduction A subset ag of a metric space (cg, d:Cg x ~ --~ R>o) is called gated (in cg) if the following holds: F o r every C e oK, there exists a C ' • d

such that d(C, D) = d(C, C') + d(C', D)

for all D • ag. Pictorially, C ' is the gate of ag with respect to C. If C" is a n o t h e r gate, then

d(C, C") = d(C, C') + d(C', C") = d(C, C") + d(C", C') +

that is,

d(C', C"),

d(C', C") = 0. Thus, the gate C ' is uniquely determined by C, and we can set

C' = :p r ~ C a n d call C ' the projection of C o n t o ag. The c o n t e n t s of this note have g r o w n out of a n analysis of Tits' projection maps

AMS (1980) subject classification:Primary 54E35. Secondary 14L35, 14L40, 05C25, 05C75, 51F15.

Manuscript received November 12, 1986, and in final form, February 13, 1987. 112

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in Coxeter complexes and buildings as defined and studied in I-T1], p. 29 ff, p. 50 f, p. 234 ft. Here, ~ is the set of "chambers" (maximal simplices) of a certain simplicial complex, and d is defined by the adjacency relation, where two chambers are adjacent if their intersection is a face of codimension 1. (Thus, d is a special case of the shortest path metric on a graph.) It is a basic fact that the stars of lower dimensional simplices in buildings are always gated. Most results of IT1] on the geometrical properties of buildings are based on the corresponding projection maps. We have observed that some of these results can be proved in the general context defined above, thereby simplifying and clarifying some of the arguments of I-TI]. After we had obtained the results of the present note, H. J. Bandelt kindly informed us of the papers [ G - W ] , [H], [I], which introduce the notion of gated sets (Chebyshev sets) in quite different contexts. [ G - W ] deals with certain optimization problems and is to be credited for the term "gated". [H] deals with so-called ternary spaces which axiomatize various "between"-relations in geometry. A special case, the so-called media, had been treated before in [I] in the context of lattice theory. None of these papers contains our main result stated below. Further comments on Tits' work will be given in §3.

2. Properties of gated sets and projection maps Let (~, d) be a metric space, where d : ~ x c~___, R>~o is a distance function on cg. For C, D e ~, define the segment [C, D] by [C, D] := {E e ~ l d(C, D) = d(C, E) + d(E, D)}. A subset ~ ' _ cg is called convex (more precisely, convex in ~ ) if, for any two points C, D e .4, the whole segment [C, D] is contained in ~¢. Notice that the whole space (g is convex, and that the intersection of any family of convex sets is convex. Therefore, the convex hull 0~ of some subset ~ ' ~_ ~ can be defined as the intersection of all convex sets containing ~'. It is clear that a segment is not necessarily convex, that is, usually the convex hull of a two-element subset {C, D} is strictly larger than the segment [C, D]. The smallest example is the metric space derived from the following graph on five points:

D

114

A N D R E A S W . M. DRESS A N D R U D O L F S C H A R L A U

AEQ. M A T H .

Here, the convex hull {C, D} is all o f ~ , but the central point is not contained in the segment [C, D]. We define the distance between two subsets d , ~ ~_ cg as usual: d(,x¢, ~ ) : = inf {d(A, B) IA e d , B E ~}. O f course, d ( C , ~ ) means d({C}, ~), for CeC~, ~ _~ cg. N o w let ~ be a gated subset of ~ as defined in the introduction. The projection C" = pr./C of an element C in particular is the unique element D e ~¢ such that d(C, ~' ) = d(C, D). The example of a proper linear subspace ~¢ of Euclidean space shows that the existence and uniqueness of a closest element in ~', for all C e ~, does not imply that o~ is gated. The following obvious result has already been observed in [H], p. 217, cf. also [ G - W ] , p. 407. PROPOSITION 1. Any gated subset o~¢ of ~ • convex in cg.

Proof Let C, D ~ ~¢ and E E r£ such that d(C, D) = d(C, E) + d(E, D). Consider E ' : = pr~jE. Then d(C, D) = d(C, E') + d(E', E) + d(D, E') + d(E', E) >~ d(C, D) + 2d(E, E'). Therefore, d(E, E') = 0, that is, E = E ' e ~¢, as desired.

[]

The reader m a y keep in mind the example of H a m m i n g space c~ = X", where X is an arbitrary finite set ("alphabet") and d(x, y) :-= I{i[xi * Yi}] for x = (xl . . . . . x,), y = (Yl . . . . . y,), xi, Yi ~ X. A segment [x, y] consists of all u such that ul e {xi, y~} for all i. Therefore, the n o n e m p t y convex sets are precisely the subsets of the form .~' = X1 × . . . x X,, where X~ _~ X are arbitrarily specified n o n e m p t y subsets. Such a subset is gated if and only if X~ = X or I X~] = 1 for all i. Indeed, if I Xil ~> 2, X~ :~ X for some i, and if x is such that x~ ¢ X~, then there are at least two elements y e d such that d(x, y) = d(x, ~¢). Therefore, ~ c a n n o t be gated. O n the other hand, if .~¢ = {y [yj = zj f o r j e J} for some subset J ~_ {1 . . . . . n} and elements z i ~ X, then for a given x, the element x' e d defined by x) = zj for j e J, x~ = x i for i ¢ J satisfies the condition d(x, u) = d(x, x') + d(x', u) for all u ~ ~ .

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W e see that in b i n a r y H a m m i n g space, I XI = 2, all convex sets are gated, whereas for I XL i> 3 there exist lots of convex subsets which are n o t gated. M o r e generally, consider the p r o d u c t space ~ = c£ 1 x . . . x ~ , of metric spaces (~i, di) (i = 1. . . . . n) with the L l - m e t r i c d((C~ . . . . . C,), (D 1. . . . . D,)) : = dl (Cx, D~) + . . . + d,(C,, D.). It is easily verified that a subset ~¢ ~_ ~ is g a t e d if a n d only if it is the p r o d u c t of gated subsets d l ~- cgi:

,~¢ = a¢ 1

x d2

x ...

x a¢..

PROPOSITION 2. (cf. [ H ] , Lemma 1.8, [I], 1.7). I f ag c cg is gated and M ~_ ag is gated in ~¢, then ~ is gated in ~, and pr.~ = pr~ o pr ~, where pr~ denotes the projection from d onto ~. Proof Let C ~ cg a n d B ~ M. Then d(C, B) = d(C, pr~,C) + d(pr¢C, B) = d(C, pr¢C) + d(prolC, pri~(pr.~lC)) + d(pr'~(pr¢C), B) = d(C, pr:~(pr.~C)) + d(pr'~(pr~,C), B). LEMMA 1. The projection map onto any gated subset d More precisely, for all C, D ~ c~ the following holds:

[] of ~ is non-expanding.

d(pr~,C, pr /D) + [d(C, p r i G ) - d(D, pr~/D) I