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Self perimeter of convex sets

Gershon Wolansky

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.

Self perimeter of convex sets

Abstract

This paper introduces a natural definition for the volume of the unit ball in -dimensional normed spaces . This definition preserves the Euclidean relation between the perimiter and the volume of the unit ball in . We show that this volume definition is invariant under origin-preserving affine transformations and polar duality. For , 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 via a recursive integration over the boundary, utilizing -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.

Paper Structure

This paper contains 18 sections, 12 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Fundamental definitions
  3. Self perimeter in the 2-plane
  4. Self-Perimeter of a Regular $k$-gon
  5. Self-Perimeter of Non-Perfect $k$-gons and Triangles
  6. Self perimeter in $\mathbb R^n$
  7. Invariance of the Self-Perimeter Under Affine Transformations
  8. Examples
  9. Uniqueness of optimal center
  10. Convexity of the Self-Perimeter Functional for an $n$-Simplex
  11. Proof of Strict Convexity
  12. Strict Convexity for Cartesian Products
  13. Base Case ($k = 2$):
  14. Inductive Hypothesis:
  15. Inductive Step:
  16. ...and 3 more sections

Key Result

Lemma 2.1

For $\theta\in S^{n-1}$ $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1: Schneider schneider2014convex
  • Lemma 3.1
  • proof
  • Theorem 3.1: golab1932martini2001
  • Corollary 3.1: Golkabgolab1932
  • Remark 3.1
  • Theorem 3.2: Makeevmakeev2003
  • proof
  • ...and 14 more