Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

Kwok-Kun Kwong, Yong Wei

TL;DR

This work develops a robust, quantitative framework for bounding the Alexandrov-Fenchel deficit on closed hypersurfaces by combining a weighted Minkowski integral formula with a Jensen deficit identity. By introducing a general integral identity that links the weighted curvature integrals to the support function, the authors produce a family of sharp upper bounds for the AF deficit that incorporate a natural distance to a reference ball, and they identify equality cases with spheres about the origin. A key contribution is the stability analysis, which expresses deficits as explicit $L^2$-distances to balls (notably the Steiner ball) and shows the AF-based inequalities can strictly improve classical isoperimetric bounds under optimal origin placement. The results extend to space forms and warped-product ambient manifolds, yielding generalized deficit–distance–integral inequalities and clear characterization of equality, thereby broadening the toolkit for quantitative geometric inequalities beyond flow-based methods.

Abstract

We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities with different weight functions. Furthermore, our estimates are more quantitative as they include a distance term measuring the domain's deviation from a reference ball. We also analyze the stability of a weighted geometric inequality from a recent paper \cite{kwong2023geometric} via analysis of the support function on the sphere and show that, with an optimal choice of the origin, this inequality is stronger than the classical isoperimetric inequality.

Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

TL;DR

This work develops a robust, quantitative framework for bounding the Alexandrov-Fenchel deficit on closed hypersurfaces by combining a weighted Minkowski integral formula with a Jensen deficit identity. By introducing a general integral identity that links the weighted curvature integrals to the support function, the authors produce a family of sharp upper bounds for the AF deficit that incorporate a natural distance to a reference ball, and they identify equality cases with spheres about the origin. A key contribution is the stability analysis, which expresses deficits as explicit -distances to balls (notably the Steiner ball) and shows the AF-based inequalities can strictly improve classical isoperimetric bounds under optimal origin placement. The results extend to space forms and warped-product ambient manifolds, yielding generalized deficit–distance–integral inequalities and clear characterization of equality, thereby broadening the toolkit for quantitative geometric inequalities beyond flow-based methods.

Abstract

We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities with different weight functions. Furthermore, our estimates are more quantitative as they include a distance term measuring the domain's deviation from a reference ball. We also analyze the stability of a weighted geometric inequality from a recent paper \cite{kwong2023geometric} via analysis of the support function on the sphere and show that, with an optimal choice of the origin, this inequality is stronger than the classical isoperimetric inequality.

Paper Structure

This paper contains 14 sections, 22 theorems, 113 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Minkowski formula
  4. Jensen deficit
  5. An integral identity involving the support function and the $k$-th mean curvature
  6. Sharp upper bounds for the Alexandrov-Fenchel deficit
  7. Results for $k$-convex hypersurfaces
  8. Results for convex hypersurfaces
  9. Stability of a weighted geometric inequality
  10. Stability of the inequality \ref{['ineq1']}
  11. Comparison with the isoperimetric inequality
  12. Generalizations
  13. The space form case
  14. The warped product case

Key Result

Theorem 1.1

Let $\Sigma=\partial \Omega$ be a smooth, closed hypersurface in $\mathbb{R}^{n+1}$ with $H_k \ge 0$. Then where $\delta_{2, k}$ is the weighted $L^2$ distance (see Section sec. identity), $T_{k}$, $A$ are the $k$-th Newton tensor and shape operator of $\Sigma$ respectively, and $X^T$ is the tangential component of the position vector $X$. In particular, we have an upper bound for the deficit in

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Theorem 3.1
  • ...and 29 more