Self perimeter of convex sets

TL;DR

This paper defines a natural volume for unit balls in n-dimensional normed spaces, explores its properties, derives explicit formulas in 2D, extends to higher dimensions, and discusses related geometric problems.

Contribution

It introduces a new volume definition for unit balls that preserves Euclidean relations and invariance properties, with explicit formulas and extensions to higher dimensions.

Findings

The volume definition is invariant under affine transformations and duality.

Explicit integral formula for the self-perimeter in 2D.

Perturbative solutions for the Alexandrov-type problem in 2D.

Abstract

This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball $B$ in $R^n$. We show that this volume definition is invariant under origin-preserving affine transformations and polar duality. For $n=2$, we derive an explicit integral formula for the self-perimeter of the unit ball, extend it to non-centrally symmetric sets;. The construction is extended to $\mathbb{R}^n$ via a recursive integration over the boundary, utilizing $(n-1)$-dimensional volumes of planar intersections. Finally, we pose and discuss an Alexandrov-type problem for the associated surface measure, providing perturbative solutions in the 2D case. In particular we prove that, generically, any perturbation of the surface measure of the Euclidean 2-D disk yields a 4-fold symmetric convex set in the leading order.