Isoperimetric relations for inner parallel bodies

TL;DR

This paper investigates the properties of inner parallel bodies of convex shapes, focusing on isoperimetric ratios and correcting previous inaccuracies by analyzing boundary structures and form bodies.

Contribution

It provides new insights into the behavior of inner parallel bodies and corrects prior errors related to their boundary properties in convex geometry.

Findings

Identifies errors in previous relations involving inner parallel bodies.

Establishes corrected results for convex bodies with specific boundary structures.

Analyzes isoperimetric and quermassintegral properties of inner parallel bodies.

Abstract

We analyze aspects of the behavior of the family of inner parallel bodies of a convex body for the isoperimetric quotient and deficit of arbitrary quermassintegrals. By means of technical boundary properties of the so-called form body of a convex body and similar constructions for inner parallel bodies, we point out an erroneous use of a relation between the latter bodies in two different works. We correct these results, limiting them to convex bodies having a very precise boundary structure.