Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The double spherical cap rearrangement of planar sets

Chiara Gambicchia

TL;DR

The paper investigates the double spherical cap rearrangement $F_v$ of planar sets $E\subset \,\,\mathbb{R}^2$, a symmetry that preserves both area and barycenter. It develops a BV-framework for the circular distribution $v_E(r)$ and the associated angle function $ heta_v(r)$, establishing that $v_E\,\in BV(0,\infty)$ and $\theta_v\in BV_{\rm loc}$, and derives precise relations between radial/perimeter measures and the slices $E_r$. The main result shows that, in dimension two, if all non-trivial circular slices of $E$ are disconnected, then the double spherical cap rearrangement does not increase the local perimeter, i.e., $P(F_v; \,\phi(B\times \mathbb{S}^1)) \le P(E; \,\phi(B\times \mathbb{S}^1))$ for any Borel $B\subset (0,\infty)$. The paper also provides a counterexample in higher dimensions demonstrating that this perimeter-decreasing property fails in general, clarifying the limitations of the rearrangement beyond the plane and linking to classical work by Bonnesen; together, these results refine understanding of symmetry-based methods in isoperimetric problems and shape optimization.

Abstract

This paper is devoted to the proof of an isoperimetric property of the double spherical cap rearrangement of planar sets under the assumption of disconnection of non-trivial spherical slices. Additionally, the higher-dimensional case is briefly discussed; in particular, an explicit counterexample is given, thus explaining why an analogous result cannot hold.

The double spherical cap rearrangement of planar sets

TL;DR

The paper investigates the double spherical cap rearrangement of planar sets , a symmetry that preserves both area and barycenter. It develops a BV-framework for the circular distribution and the associated angle function , establishing that and , and derives precise relations between radial/perimeter measures and the slices . The main result shows that, in dimension two, if all non-trivial circular slices of are disconnected, then the double spherical cap rearrangement does not increase the local perimeter, i.e., for any Borel . The paper also provides a counterexample in higher dimensions demonstrating that this perimeter-decreasing property fails in general, clarifying the limitations of the rearrangement beyond the plane and linking to classical work by Bonnesen; together, these results refine understanding of symmetry-based methods in isoperimetric problems and shape optimization.

Abstract

This paper is devoted to the proof of an isoperimetric property of the double spherical cap rearrangement of planar sets under the assumption of disconnection of non-trivial spherical slices. Additionally, the higher-dimensional case is briefly discussed; in particular, an explicit counterexample is given, thus explaining why an analogous result cannot hold.

Paper Structure

This paper contains 4 sections, 8 theorems, 107 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Technical results
  4. Proof of Theorem \ref{['thm sph rearr']}

Key Result

Theorem 1.4

Let $E$ be a set in $\mathbb{R}^2$ with finite perimeter and finite volume and let $v:(0, +\infty)\to [0,+\infty)$ be its circular distribution. Then the following hold: Moreover, defining $\Gamma_E$ as in def Gamma_E, one has for any Borel set $B\subseteq (0,+\infty)$. In particular, if all non-trivial slices of $E$ are disconnected, then it holds for any Borel set $B\subseteq (0,+\infty)$.

Figures (5)

  • Figure 1: The sets $D_\theta(r)$ and $S_\theta(r)$.
  • Figure 2: A set and its double spherical cap rearrangement.
  • Figure 3: The example showing that, in general, the double spherical cap symmetrization does not decrease the perimeter.
  • Figure 4: Decomposition of $\nu^E$ as sum of its radial and tangential components.
  • Figure 5: The example showing that, in general, the double spherical cap symmetrization does not decrease the perimeter in dimension higher than $2$.

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1: Leibniz rule in BV
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • ...and 7 more