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Alexandrov's Soap Bubble Theorem for Polygons

Marco Bonacini, Riccardo Cristoferi, Ihsan Topaloglu

Abstract

Regular polygons are characterized as area-constrained critical points of the perimeter functional with respect to particular families of perturbations in the class of polygons with a fixed number of sides. We also review recent results in the literature involving other shape functionals as well as further open problems.

Alexandrov's Soap Bubble Theorem for Polygons

Abstract

Regular polygons are characterized as area-constrained critical points of the perimeter functional with respect to particular families of perturbations in the class of polygons with a fixed number of sides. We also review recent results in the literature involving other shape functionals as well as further open problems.

Paper Structure

This paper contains 4 sections, 2 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Criticality Conditions
  3. Alexandrov's Theorem for Polygons
  4. Further Results and Conjectures

Key Result

Theorem \oldthetheorem

A polygon $\mathcal{P}\in\mathscr{P}_N$ is stationary with respect to the sliding variation as in Definition def:sliding on the $i$-th side, for $i\in\{1,\ldots,N\}$, under area constraint if and only if where $\psi$ is defined in eq:psi. A polygon $\mathcal{P}\in\mathscr{P}_N$ is stationary with respect to the tilting variation as in Definition def:tilting on the $i$-th side, for $i\in\{1,\ldots

Figures (4)

  • Figure 1: Notation used in the statement of the Main Result depicting the angles $\theta_i$, $\theta_{i+1}$, $\alpha_i^-$, $\alpha_i^+$, and the length $\ell_i$ of the side $\mkern2mu \overline{\mkern-2mu P_i P_{i+1}}$.
  • Figure 2: A polygon $\mathcal{P}$ and its variation $\mathcal{P}_t$ (shaded region) as in Definition \ref{['def:sliding']}, obtained by sliding the side $\mkern2mu \overline{\mkern-2mu P_i P_{i+1}}$ in the normal direction at a distance $t>0$.
  • Figure 3: A polygon $\mathcal{P}$ and its variation $\mathcal{P}_t$ (shaded region) as in Definition \ref{['def:tilting']}, obtained by tilting the side $\mkern2mu \overline{\mkern-2mu P_i P_{i+1}}$ around its midpoint $M_i$ by an angle $t>0$.
  • Figure 4: A polygon $\mathcal{P}$ and its variation $\mathcal{P}_t$ (shaded region) as in Definition \ref{['def:ricIvar']}, obtained by moving the vertex $P_i$ parallel to the diagonal $\mkern2mu \overline{\mkern-2mu P_{i-1} P_{i+1}}$ at a distance $t>0$.

Theorems & Definitions (11)

  • Definition \oldthetheorem: Sliding of one side
  • Definition \oldthetheorem: Tilting of one side
  • Definition \oldthetheorem: Moving of one vertex
  • Definition \oldthetheorem: Stationarity
  • Theorem \oldthetheorem: Stationarity conditions
  • proof
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Alexandrov's Theorem for polygons
  • proof
  • Remark \oldthetheorem
  • ...and 1 more