The Characteristic Properties of the Minimal Lp-Mean Width

Giannopoulos proved that a smooth convex body K has minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn-1, is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal Lp-mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal Lp-mean width of convex bodies and prove the existence and uniqueness of the minimal Lp-mean width in its SL(n) images. In addition, we establish a characterization of the minimal Lp-mean width, conclude the average Mp(K) with a variation of the minimal Lp-mean width position, and give the condition for the minimum position of Mp(K).

A SPHERICAL MAPPING AND BORSUK CONJECTURE IN RIEMANNIAN AND NON-EUCLIDEAN SPACES

We introduce an analog of the spherical mapping for convex bodies in a Riemannian $n$-manifold, and then use this construction to prove the Borsuk conjecture for some types of such bodies. The Borsuk conjecture is that each bounded set $X$ in the Euclidean $n$-space can be covered by $n +1$ sets of smaller diameter. The conjecture was disproved recently by Kahn and Kalai. However Hadwiger proved the Borsuk conjecture under the additional assumption that the set $X$ is a smooth convex body. Here we extend this result to convex bodies in Riemannian manifolds under some further restrictions.

A Note on the Borsuk Conjecture

According to the still unproved conjecture of Borsuk [1] a bounded subset A of the Euclidean n-space En is a union of n + 1 sets of diameters less than the diameter D of A. Since A can be imbedded in a set of constant width D, [2], it may be assumed that A is already of constant width. If in addition A is smooth, i. e., if through every point of its boundary ∂A there passes one and only one support plane of A, then the truth of Borsuk′s conjecture can be proved very easily [3]. The question arises whether Borsuk′s conjecture holds also for arbitrary smooth convex bodies, not merely for those of constant width. Since it is not known whether a smooth convex body K can be imbedded in a smooth set of constant width D, the answer is not immediate. In this note we show that the answer is affirmative.