On Properties of $p$-Critical Points of Convex Bodies

Abstract

Properties of the $p$-measures of asymmetry and the corresponding affine equivariant $p$-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of $p$-critical points with respect to $p$ on $(1, +∞)$ is confirmed, and the connections between general $p$-critical points and the Minkowski-critical points ($∞$-critical points) are investigated. The behavior of $p$-critical points of convex bodies approximating a convex bodies is studied as well.