On the total perimeter of disjoint convex bodies

TL;DR

This paper introduces a pseudometric for convex planar curves and uses it to provide a concise proof of Pinchasi's theorem, which bounds the total perimeter of disjoint convex bodies within a convex set.

Contribution

The paper presents a new pseudometric on convex curves and applies it to simplify the proof of a perimeter sum bound for disjoint convex bodies.

Findings

The pseudometric has well-defined basic properties.

The total perimeter of disjoint convex bodies is bounded by a function of the containing set.

A concise proof of Pinchasi's theorem is provided.

Abstract

In this note we introduce a pseudometric on convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a short proof of the theorem by Pinchasi that the sum of perimeters of $k$ convex planar bodies with disjoint interiors contained in a convex body of perimeter $p$ and diameter $d$ is not greater than $p+2(k-1)d$.