Communications in Analysis and Geometry

Volume 30 (2022)

Number 10

Isoperimetric relations for inner parallel bodies

Pages: 2267 – 2283



M. A. Hernández Cifre (Departamento de Matemáticas, Universidad de Murcia, Spain)

E. Saorín Gómez (ALTA Institute for Algebra, Geometry, Topology and their Applications, Universität Bremen, Germany)


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.

This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. The work is supported by PID2021-124157NB-I00, funded by MCIN/ AEI/10.13039/501100011033/ “ERDF A way of making Europe”, and Fundación Séneca projects 19901/GERM/15 and 21899/PI/22.

Received 22 October 2019

Accepted 15 June 2020

Published 29 September 2023