PETER M. G R U B E R

T H E S P A C E O F C O M P A C T S U B S E T S O F E a

ABSTRACT. Let Yd" be the space of compact subsets of E a, endowed with the Hausdorff- metric. It is shown that the isometries of ~ onto itself are the mappings generated by rigid motions of E a.

1. INTRODUCTION

The following paper contains information on the isometries of the space of compact subsets of the d-dimensional Euclidean space E a.

For subsets C, D of E a and A ~ ~ define C + D := {x + y I x ~ C, y ~ D} and AC := {Ax I x ~ C}. Let B denote the solid unit ball of E a. Define

~(C,D):=inf{A~fl~ + [ C c D + A B , D c C + A B } forC, D ~ .

is a common metric for ~ , called the Hausdorff-metric (cf. [3, p. 90] and [5]).

Recently Schneider [4] raised the problem of describing the isometries of onto itself. In §2 it will be shown that these isometries are the maps I of into itself, generated by rigid motions i of Ea; that is, I ( C ) = i(C) (:={i(x) l x ~ C}) for each C ~ . Corresponding results for the space of convex bodies and for the so-called convex-ring were given by Schneider [4] and Gruber [1].

bd, int and dim denote boundary, interior and dimension. Let II I/be the Euclidean norm. For p ~ E a we shall write simply p instead of {p}.

2. THE ISOMETRIES OF ft,(" ONTO ITSELF

THEOREM. The isometries of Jd onto itself are exactly those maps of into itself, generated by isometries o r E d.

Proof. It is obvious that each isometry of E a generates an isometry of ~f" onto itself.

Suppose now that I is an isometry of ~'- onto itself. In our proof we shall make use of the following result of Gruber [1]:

(1) The isometries of ~ into itself which map some point of E a onto a point of E a are precisely those generated by isometries of E a.

Our first goal is to show that

(2) p , q ~ E a , p ¢ q :~I(p) c bd(I(q) + lip - qll B).

Let us suppose the contrary: i.e. I(p) ¢ bd(I(p) + lip - qll B ) . W e have 3(l(p), l (p)) = 3(p, q) = l ip - ql] . This, together with the compactness of

Geometriae Dedicata 9 (1980) 87-90. 0046-5755/80/0091-0087500.60 Copyright © 1980 by D. Reidel Publishing Co., Dordrecht, Holland and Boston, U.S.A.

88 rEfER M. GRUBER

I (p) and I(q), implies I (p) c I(q) + liP - ql[ B. Hence I (p) n int(I(q) + lip - ql[ B) 4 ~. From this it follows that

(3) C : - - - ( I ( p ) + hB) n ( I (q ) + hB) e ~ i n t C 4

for h := lip - q[[/2. Obviously

(4) C c I(p) + AB, C ~ I(q) + ~B.

On the other hand, we have

(5) I(p) c C + hB, X(q) c C + ;~B.

Indeed, S(t(p), I((p + q)/2)) = ~(i((p + q) /2) , i(q)) = lip - qll/2 = ~. Con- sequently,

(6) I((p + q)/2) c I(p) + hB, I(q) + hB,

(7) I(p), I(q) = I((p + q)/2) + aB.

(3) and (6) imply I((p + q)/2) c C. This, together with (7), yields (5). From (4) and (5) we conclude 3(I(p), C), 3(C, I(q))<~ ,~. Since 3(I(p), I (q) )= l ip - q]l = 2a the triangle inequality for 3 finally shows that

(8) ~(X(p), c ) = ~(c , t(q)) = ~.

By (3) int C 4- ~. We remove from C a nonempty open subset contained in int C, whose diameter is < ,k This gives a set D. Obviously

(9) BevY', D ~ C

and D + ~,B = C + ,~B. This combined with (4) and (5) shows that 8(I(p), D), 8(D, I(q)) <~ L From this, together with the triangle inequality for 8, we conclude

(10) ~(I(p), D) = ~(D, X(q)) = a.

Since I is an isometry of ~ e~ onto itself, the same holds true for I-1. Therefore (8) and (10) imply 8(p, I - 1(C)) = 8(I- 1(C), q) = 8(p, I - I(D)) = 8(I- I(D), q) = A = Hp - q[[/2. Hence ~ 4- I-1(C), I - I (D) c (p + AB) c~ (q + AB) = (p + q)/2. Consequently, I -1(C) = I - I (D) (= (p + q)/2) and thus C = D, contradicting (9). This proves (2).

(2) and the fact that I(p) is compact for each p e E a imply

(11) p, q ~ E a, p # q ~ for each x e I(p) there exists a point y e I(q) such that HP - qH = n x - yH,

(12) p, q e E a, p ~ q :~ for all u e I(p), y e I(q) we have ItP - qH Ilu - y l l .

THE SPACE OF C O M P A C T SUBSETS OF E a 89

We will show by induction the following simple proposition:

(13) Let S and S' be spheres of equal radius with dim S = dim S' and f : S ~ S ' a map which does not decrease distances. Thenf is onto.

(13) obviously holds if dim S = dim S' = 0. Let n ~ I~ and suppose that (13) holds if dim S = dim S' = n. Now assume that dim S = dim S' = n + 1. For antipodal points p, q E S denote by C(p, q) the great circle of S which is equidistant from p and q. Define C'(p', q') analogously. We first prove the following:

(14) p, q E S antipodal =- p' := f (p ) , q' := f (q ) ~ S ' are antipodal and f maps C(p, q) onto C'(p' , q').

Since p, q are antipodal and f does not decrease distances, p', q' are also antipodal. Since f maps p, q onto p', q' and f does not decrease distances, C(p, q) is mapped into C'(p' , q'). Now the induction hypothesis shows that f actually maps C(p, q) onto C'(p' , q'). This proves (14). We now assert

(15) x' ~ S ' => x' = f ( x ) for suitable x E S.

To prove (15), fix a pair of antipodal points p, q ~ S and choose antipodal points r', s' ~ C'(p' , q') such that x' ~ C'(r', s ' ) . fmaps C(p, q) onto C'(p' , q') by (14). Therefore there is a point r ~ C(p ,q ) such that r ' = f ( r ) . Let s E C(p, q) be the antipode of r. Sincefdoes not decrease distances, the image of s is the antipode of the image of r, i.e., s' = f ( s ) . f maps C(r, s) onto C'(s', r') by (14). Hence there is a point x ~ C(r, s) c S such that x' = f ( x ) . This proves (15). (15) shows tha t f i s onto, concluding the induction and thus proving (13).

Finally we show,

(16) p ~ E a :~ I (p) consists of one point only.

Suppose that for somep ~ E a there exist u, x ~ I(p), u v ~ x. Let A := [Ix - u[[. (11) implies that for each q ~p + h bd B there exists a point y(q) E I(q) such that IIx- y(q)l] = lip- qll--~. Hence, the mapping q-+y(q) maps the sphere p + h bd B into the sphere x + h bd B. Because of (12) it does not decrease distances. This, together with (13), implies that this mapping is onto. Since u E x + A bd B, there exists a point q ~p + h bd B such that y(q) = u. Hence Irp - q[I = A > 0 = Hu - y(q)l[, contradicting (12). This proves (16).

It follows from (1) and (16) that I is generated by an isometry of E a. This concludes the proof of our theorem.

3. F I N A L REMARKS

The question of giving a complete description of the isometrics of Jg" into itself is still open. We conjecture that every isometry of ~ into itself actually is onto. If one imposes different metrics on ~ , there then arises the problem of

90 PETER M. GRUBER

describing the corresponding isometrics. Sometimes there might be a close connection with ' isometric' mappings of the underlying (linear) space (cf. [2]).

ACKNOWLEDGEMENTS

This paper was written during a stay in April 1977 at the Mathematical Institute of the Hungarian Academy of Sciences in Budapest for which I express my gratitude to Professor L. Fejes T6th. I should also like to thank the referee for his remarks which prevented me from publishing a known result.

B I B L I O G R A P H Y

1. Gruber, P. M., ' Isometrien des Konvexringes', Coil. Math. (in print). 2. Gruber, P. M., 'Isometrics of the Space of Convex Bodies of E a' Mathematika 25,

270-278 (1978). 3. Rogers, C. A., HausdorffMeasures, Cambridge University Press, 1970. 4. Schneider, R., 'Isometrien des Raumes der konvexen KOrper', Coll. Math. 33,

219-224 (1975), 5. Shephard, G.C. and Webster, R., ' Metrics for Sets of Convex Bodies', Mathematika

12, 73-88 (1965).

Author's address:

Peter M. Gruber, Institut fiir Analysis, Technische Universit~t, A-1040 Wien, Guflhausstrafle 27 Austria

(Received January 20, 1978)