Hardy Type Inequalities on Domains with Convex Complement and Uncertainty Principle of Heisenberg

Abstract

We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. We assume that the complement of the subdomain is a non-empty convex set. We prove an extension of the Hadwiger theorems about approximations of convex compact sets by polytopes and obtain some generalizations and improvements of several Hardy type multidimensional inequalities. In particular, in the last section we present an improvement of a two-dimensional inequality, connected with the uncertainty principle of Heisenberg.