H. Radstrom, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165--169.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 log(1 Gamma p) Gamma jj 1 ent( Then, for any q 1, dH (W oe ; W ) 0 : Remark. For p close to 1, the diameter of these Wulff crystals is of order Gamma ln(1 Gamma p) Proof. The functions and oe are the support functions of the crystals W and W oe . By an identity due to Radstrom [26] and Hormander [20] we have dH (W oe ; W ) sup 2S joe( p) Gamma ( p)j : The result follows then directly from the uniformity of the asymptotic expansion of Theorem 1:1. To determine W oe , we need to find for each direction 2 S r( min w2S Delta w Then r( will be the ....

H. Radstrom, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165--169.

.... tools used for analysing set valued functions include the support function technique for describing convex compact sets (see e.g. 10] and methods of embedding the cone of convex compact subsets of R n in a linear normed space, with addition defined as the Minkowski sum of sets ( 2] 7] 8] [9]) In any such linear normed space, we introduce a partial order generated by the set inclusion order in the cone of convex compact sets. Thus a multifunction with convex compact images from R to R n is considered as an abstract function with values in a partially ordered normed linear space. ....

.... ffi (B 2 ; Delta) ffi (A 2 ; Delta) B 2 A 2 : There are various ways to construct a linear normed vector space D(R n ) relative to the Minkowski sum, in which the cone C(R n ) is embedded by an embedding J : C(R n ) D(R n ) with the following properties(see e.g. 7] [9], 8] i) J(A B) J(A) J(B) ii) J(A) J(A) 0: iii) J(A) J(B) A = B: iv) kJ(A) Gamma J(B)k = haus(A; B) These properties imply J(f0g) where is the zero element of D(R n ) A simple embedding is J(A) ffi (A; Delta) It is easy to check , using the above stated ....

H. Radstrom,An embedding theorem for spaces of convex sets, Proceedings of the American Mathematical Society 3 (1952) 165--169.

.... Omega coXdP: Lemma 2.3. Debreu, 1966) A random set X : Omega C(E) has E(coX) 2 K(E) if R Omega jjcoX jjdP 1. In obtaining laws of large numbers for the Banach space valued random sets, many authors have found it useful to embed the metric space K(E) into a part of a Banach space. Radstrom (1952) showed that the collection of compact convex subsets of a Banach space can be isometrically embedded as a convex cone in a normed space. The Radstrom embedding technique was refined by Gine, Hahn and Zinn (1983) who showed that the isometric embedding of K(E) could be into the support functions ....

Radstrom, H. (1952). An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc., 3 (1952), 165-169.

....directly the theory of the Bochner integral to compute the expected value of a random variable on (C; B C ) as done in section 2, since that theory is constructed for Banach spaces and C does not have a linear structure. Nevertheless, as pointed out by Hiai and Umegaki [14] see also Radstrom [27]) C can be embedded as a convex cone in a Banach space, and this suffices to allow the theory of Bochner integration to be applied to C valued functions. As with the space (H (IR m ) BH ) we say that a function Upsilon : Omega C is simple if there exist sets C 1 ; CN 2 C and E ....

H. Radstrom, "An embedding theorem for spaces of convex sets", Proc. Amer. Math. Soc., 3, 165-169 (1952 ).

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