# On a reverse isoperimetric inequality for relative outer parallel bodies

## TL;DR

This paper establishes a reverse isoperimetric inequality for relative outer parallel bodies with respect to a convex body, providing conditions for equality and exploring related inequalities through convexity properties.

## Contribution

It introduces a novel reverse isoperimetric inequality for relative outer parallel bodies and analyzes equality conditions and related inequalities using quermassintegrals.

## Key findings

- Reverse isoperimetric inequality proved for relative outer parallel bodies.
- Equality conditions characterized for the inequality.
- Additional inequalities derived from convexity of quermassintegrals.

## Abstract

We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body $E$, along with its equality condition. Based on the convexity of the sequence of quermassintegrals of Minkowski sums we also prove further inequalities.