Let K and L be compact convex sets in Rn. The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ K having at most n+ 1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 ≤ d ≤ n − 1. It is also shown that the following two statements are equivalent: (i) For every polytope Q ⊆ K having at most d+ 1 vertices, L contains a translate of Q. (i...

Let (A,LA) be a quantum metric space. Then clearly S(A) with the metric ρLA is a compact balanced convex metric space, i.e. the metric ρLA is convex and balanced. Another important property of S(A) is that the R−valued affine continuous functions Af(S(A)) separate the points of S(A). Conversly, let (X, ρ) be a compact balanced convex metric space on which Af(X) separate the points of X. Then Af...

Let K and L be compact convex sets in Rn. Suppose that, for a given dimension 1 ≤ d ≤ n − 1, every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K. Does it follow that the original set L contains a translate of K? In other words, if K can be translated to “hide behind” L from any perspective, does it follow that K can “hide inside” L? A compact...

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we resolve the conjecture of Vergne in this non-compact setting. 0. Introduction The famous geometric quantization conjecture of Guillemin and Sternberg [9] states that for a compact pre-quantizable...