In this article we study metric spaces which admit polynomial diameter-volume inequalities for k-dimensional cycles. These generalize the notion of cone type inequalities introduced by M. Gromov in his seminal paper Filling Riemannian manifolds. In a first part we prove a polynomial isoperimetric inequality for k-cycles in such spaces, generalizing Gromov’s isoperimetric inequality of Euclidean...

It is a known approach to translate propositional formulas into systems of polynomial inequalities and consider proof systems for the latter. The well-studied proof systems of this type are the Cutting Plane proof system (CP) utilizing linear inequalities and the Lovász– Schrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS of LS that operate on polynomial ine...

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient imp...