On convex bodies of constant width
Abstract We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in R n is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.
On Reduced Bodies, Regularity of Norms, and Related Antipodality Properties
The thickness (or minimum width) of a convex body K in Euclidean space E, d ≥ 2, is the minimal distance between two distinct parallel supporting hyperplanes of K. The body K is said to be reduced if no proper convex subset of K has the same thickness. It is natural to study the notions of thickness and reduced body also in Minkowski spaces (that is, in real normed linear spaces of finite dimension). We will prove various basic results on reduced polytopes in a d-dimensional Minkowski spaceM, where the degree of regularity, that the boundary of the unit ball of M has, plays an essential role. If P is a reduced polytope in a Minkowski space M whose unit ball is m-regular, say, then every vertex of P is strictly antipodal to some (m+1)-face of P , and this statement can be extended to a characterization of reduced bodies. With the help of this general property we obtain the following further results: in a Minkowski space M with Csmooth (i.e., (d− 2)-regular in our terms) unit ball there are no reduced polytopes with d+2 vertices or d+2 facets, while any simple reduced polytope inM has to be a simplex; for d = 2 the unit ball of M is C-smooth if and only if no reduced quadrilateral exists; if the unit ball of M is C-smooth, then reduced polygons in M necessarily have an odd number of vertices. We present also new results (used as lemmas for the main statement) in the spirit of classical convexity. These refer to the boundary structure of general convex bodies and, in particular, to strict antipodality of faces of polytopes in E, but also to cross-section measures in Minkowski spaces. For example, we prove the following surprising affine result: no d-dimensional convex polytope with n vertices is strictly vertex-facet antipodal if and only if n = d + 2. Mathematics Subject Classification (AMS 2000): 52A21, 52A20, 52B12, 46B20
Some open mapping theorems for measures
Given a compact Hausdorff space X, let C(X) be the Banach space of continuous real valued functions on X with sup norm and let M(X) be its dual considered as finite regular Borel measures on X. Let U(X) denote the closed unit ball of M(X) and let P(X) denote the nonnegative measures in M(X) of norm 1. A continuous map w of X onto another compact Hausdorff space Y induces a natural linear transformation 7T of M(X) onto M( Y) defined by setting 7([k)(g) = ,(g o q) for Z E M(X) and g E C(Y). It is shown that v is norm open on U(X) and on A P(X) for any subset A of the real numbers. If 9 is open, then X is weak* open on A P(X). Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.