Improving bounds for the Perel'man-Pukhov quotient for inner and outer radii

TL;DR

This paper improves upper bounds for the ratio of successive inner and outer radii of convex bodies, generalizing classical results and introducing a new technique linking sections and projections.

Contribution

It provides tighter bounds for the Perel'man-Pukhov quotient and introduces an innovative method relating sections and projections of convex bodies.

Findings

Established improved upper bounds for the ratio of successive radii.

Generalized classical geometric inequalities by Perel'man and Pukhov.

Developed a novel technique connecting sections and projections.

Abstract

In this work we study upper bounds for the ratio of successive inner and outer radii of a convex body K. This problem was studied by Perel'man and Pukhov and it is a natural generalization of the classical results of Jung and Steinhagen. We also introduce a technique which relates sections and projections of a convex body in an optimal way.