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Some isoperimetric inequalities involving the boundary momentum

Domenico Angelo La Manna, Rossano Sannipoli

TL;DR

This work advances weighted isoperimetric theory by linking boundary momentum and curvature through two main strands: (i) sharp bounds and a quantitative stability framework for curvature-weighted functionals $\mathcal{H}(E)$ and $\mathcal{G}_{\beta}(E)$, revealing dimension-dependent threshold phenomena where balls are minimizers or maximizers, with precise 2D thresholds at $\beta=1$ and $\beta=5/3$, and a curvature-centroid construction for equality cases; (ii) a thorough study of boundary momentum optimization, proving a sharp planar bound for undecomposable sets and introducing a scaling-invariant functional $\mathcal{F}(E)$ in higher dimensions, for which maximizers exist in the convex class and the ball is the unique maximizer among nearly spherical sets, accompanied by quantitative stability results. The results sharpen Aleksandrov–Fenchel-type inequalities in this weighted setting, provide new planar refinements of isoperimetric-type bounds, and establish constructive methods for identifying extremal shapes under combined curvature and momentum constraints. The findings have implications for shape optimization with density weights and deepen the understanding of how geometric quantities control extremal configurations in multi-parameter isoperimetric problems.

Abstract

The aim of this paper is twofold. In the first part we focus on a functional involving a weighted curvature integral and the quermassintegrals. We prove upper and lower bounds for this functional in the class of convex sets, which provide a stronger form of the classical Aleksandrov-Fenchel inequality involving the $(n-1)$ and $(n-2)$-quermassintegrals, and consequently a stronger form of the classical isoperimetric inequality in the planar case. Moreover, quantitative estimates are proved. In the second part we deal with a shape optimization problem for a functional involving the boundary momentum. It is known that in dimension two the ball is a maximizer among simply connected sets when the perimeter and centroid is fixed. We show that the result still holds in the class of undecomposable sets. In higher dimensions the same result does not hold and we consider a new scaling invariant functional that might be a good candidate to generalize the planar case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension.

Some isoperimetric inequalities involving the boundary momentum

TL;DR

This work advances weighted isoperimetric theory by linking boundary momentum and curvature through two main strands: (i) sharp bounds and a quantitative stability framework for curvature-weighted functionals and , revealing dimension-dependent threshold phenomena where balls are minimizers or maximizers, with precise 2D thresholds at and , and a curvature-centroid construction for equality cases; (ii) a thorough study of boundary momentum optimization, proving a sharp planar bound for undecomposable sets and introducing a scaling-invariant functional in higher dimensions, for which maximizers exist in the convex class and the ball is the unique maximizer among nearly spherical sets, accompanied by quantitative stability results. The results sharpen Aleksandrov–Fenchel-type inequalities in this weighted setting, provide new planar refinements of isoperimetric-type bounds, and establish constructive methods for identifying extremal shapes under combined curvature and momentum constraints. The findings have implications for shape optimization with density weights and deepen the understanding of how geometric quantities control extremal configurations in multi-parameter isoperimetric problems.

Abstract

The aim of this paper is twofold. In the first part we focus on a functional involving a weighted curvature integral and the quermassintegrals. We prove upper and lower bounds for this functional in the class of convex sets, which provide a stronger form of the classical Aleksandrov-Fenchel inequality involving the and -quermassintegrals, and consequently a stronger form of the classical isoperimetric inequality in the planar case. Moreover, quantitative estimates are proved. In the second part we deal with a shape optimization problem for a functional involving the boundary momentum. It is known that in dimension two the ball is a maximizer among simply connected sets when the perimeter and centroid is fixed. We show that the result still holds in the class of undecomposable sets. In higher dimensions the same result does not hold and we consider a new scaling invariant functional that might be a good candidate to generalize the planar case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension.
Paper Structure (16 sections, 22 theorems, 178 equations, 1 figure)

This paper contains 16 sections, 22 theorems, 178 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary results and Definitions
  3. Basic notions and definitions
  4. Hausdorff distance and curvature measure
  5. Nearly spherical sets
  6. Some basic notions on convex bodies
  7. Support functions and curvatures of uniformly convex sets
  8. Quermassintegrals: definition and properties
  9. Isoperimetric inequalities involving curvature and boundary momentum
  10. Lower bounds on the weigthed curvature
  11. Upper bound for the weighted curvature
  12. Proof of Theorem \ref{['prop:weighted1']} and consequences.
  13. Special cases: two and three dimension
  14. Quantitative version
  15. Boundary momentum inequalities
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $E\subset \mathbb{R}^n$ be an open, bounded convex set. If $0\le\beta\le n-1$, then we have If $\beta \geq \beta(n):=(n-1) (1+\tfrac{n}{n+1})$ Moreover, equality holds if and only if $E=B_r(x_0)$ for some $x_0\in \mathbb{R}^n$ and $r>0$.

Figures (1)

  • Figure 4.1: The set $E$ (in grey) is the set difference between the union of a finite number of open disjoint polygons $E_i$, for which each of them touches at least another one, and the union of a finite number of open disjoint polygons $F_j$, which are contained in $\bigcup E_i$ and must touch its boundary in a finite number of points.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • ...and 35 more