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

Alexandrov Curvature of Convex Hypersurfaces in Hilbert Space Forms

It is shown that convex hypersurfaces in Hilbert space forms have corresponding lower bounds on Alexandrov curvature. This extends earlier work of Buyalo, Alexander, Kapovitch, and Petrunin for convex hypersurfaces in Riemannian manifolds of finite dimension.

A Form of Alexandrov - Fenchel Inequalitypengfei Guan

On a relative Alexandrov-Fenchel inequality for convex bodies in Euclidean spaces

In this note we prove a localized form of Alexandrov-Fenchel inequality for convex bodies, i.e. we prove a class of isoperimetric inequalities in a ball involving Federer curvature measures. 1991 Mathematics Subject Classification: 52A20, 52A39, 52A40, 49Q15.

On Locally Convex Hypersurfaces with Boundary

In this paper we give some geometric characterizations of locally convex hypersurfaces. In particular, we prove that for a given locally convex hypersurface M with boundary, there exists r > 0 depending only on the diameter of M and the principal curvatures of M on its boundary, such that the r-neighbourhood of any given point on M is convex. As an application we prove an existence theorem for ...

On Aleksandrov-Fenchel Inequalities for k-Convex Domains

These are the lecture notes based on a course which the first named author has given at the second congress organized by the Riemann International School of Mathematics in Verbania, Italy from September 26 to October 1, 2010. The topic of the school is Nonlinear Analysis and Nonlinear PDE, with Louis Nirenberg served as the Director of the school for the year. The first named author has greatly...