Article,
Local and Global Stability of the - Curvature
CENTRAL ASIAN JOURNAL OF THEORETICAL AND APPLIED SCIENCE, 4 (5): 147-162 (May 2023)
Abstract
Origin-centered balls only, when , and only for balls when is the curvature of a smooth, strictly convex body in in known to be constant. Only for origin-symmetric ellipsoids does the --curvature remain constant if . Using the global stability result from 5, we demonstrate that for 0, the volume symmetric difference between K and a translation of the unit ball B is nearly zero if the -curvature is approximately constant. Here, we have K shrunk to the same volume of a unit ball, denoted by K. We demonstrate a comparable result for in the -distance class of origin-symmetric entities. We also demonstrate a local stability conclusion for : Any strictly convex body with 'nearly' constant curvature is 'almost' the unit ball, and this neighborhood surrounds the unit ball. Both a global stability result in R2 for and a local stability result for in the Banach-Mazur distance are demonstrated.
