Illumination and visibility problems in terms of closure operators.
Hedgehogs are a natural generalization of convex bodies of class C+2. After recalling some basic facts concerning this generalization, we use the notion of index to study differential and integral geometries of hedgehogs.As applications, we prove a particular case of the Tennis Ball Theorem and a property of normals to a plane convex body of constant width.
We establish some inequalities for general width-integrals of Blaschke-Minkowski homomorphisms. As applications, inequalities for width-integrals of projection bodies are derived.
We establish Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms, which in special cases yield some new results for intersection bodies. Moreover, we obtain two monotonicity inequalities for radial Blaschke-Minkowski homomorphisms.
We extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in , p > -2.