Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

2018/11/14 by Julian Scheuer, Guofang Wang, Chao Xia · 29 citations
Mathematics · #Ball (mathematics) #Boundary (topology) #Convex body #Convex hull #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Isoperimetric inequality #Mathematical analysis #Mathematics #Mean curvature #Point processes and geometric inequalities #Pure mathematics #Regular polygon #Unit sphere #math.AP #math.DG

paper · pdf · doi:10.4310/jdg/1645207496

published in Journal of Differential Geometry 120(2) (Lehigh University) · 21 pages. Comments welcome

arxiv created 2018/11/14 · openalex publication_date 2022/02/01 · arxiv updated 2022/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the (n+1)-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for n=2 we obtain a Minkowski-type inequality and for n=3 we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

Citations

• Embedded minimal tori in S3 and the Lawson conjecture
• A volume preserving flow and the isoperimetric problem in warped product spaces
• Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space
• Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

Cited by

• Parabolic approaches to curvature equations
• The Minkowski inequality in de Sitter space
• 2019-20 MATRIX Annals