Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball
Julian Scheuer, Guofang Wang, Chao Xia
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.
Topics & Concepts
MathematicsIsoperimetric inequalityMathematical analysisRegular polygonBall (mathematics)Unit sphereConvex bodyMean curvatureBoundary (topology)Pure mathematicsCurvatureGeometryConvex hullGeometric Analysis and Curvature FlowsPoint processes and geometric inequalitiesGeometry and complex manifolds