50 years sets with positive reach -- a survey.
A geometric proof of some inequalities involving mixed volumes.
A stability estimate for the Aleksandrow-Fnechel inequality, with an application to mean curvature.
Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes
We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex...
Around the Rogers-Shepard inequality.
Determination of compact subsets from functionals on them.
Dual affine isoperimetric inequalities.
Global geometry under isotropic Brownian flows.
Inequalities for dual affine quermassintegrals.
Intrinsic Volumes and Lattice Points of Crosspolytopes.
Mean Projections and Finite Packings of Convex Bodies.
Notes on the roots of Steiner polynomials.
On Hadwiger's problem on inner parallel bodies
We consider the problem of classifying the convex bodies in the 3-dimensional space depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner/outer parallel bodies. It turns out that this problem is closely related to some behavior of the roots of the 3-dimensional Steiner polynomial.
On star duality of mixed intersection bodies.
On the Aleksandrov-Fenchel Inequality and the Stability of the Sphere.
On the general Brunn-Minkowski theorem.
On weighted parallel volumes.
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in -dimensional Euclidean space . It is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, covariant, and associative if and only if it is addition for some . It is also demonstrated that if ,...
Qualitative and quantitative results for sets of singular points of convex bodies.
Rigidity and the Alexandrov-Fenchel Inequality.