Metric invariants of simplicial polytopesMaksym FedorchukMIT
March 10,

ABSTRACT


We shall discuss the construction problem for convex polytopes. Given the combinatorial structure of a simplicial polytope, there is an open condition on the lengths of its edges, such that there exist a polytope with edges of given lengths. Aleksandrov's theorem gives sufficient conditions for such a construction. The problem of explicitly constructing a polytope in Euclidean space is important and very difficult. From geometric point of view, it can be reduced to finding lengths of polytope's diagonals. Algebraic relations between lengths of edges and lengths of diagonals (or any other metric invariant) of simplicial polyhedron is the principal tool in our study. Their existence is also the main result of our paper. As an example of our approach, we prove the conjecture of Robbins on the degrees of generalized Heron polynomials for an inscribed ngon. This is a joint work with Igor Pak. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

