Extremum problems for the cone volume functional of convex polytopes

摘要

Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in R~n containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R~2 and R~3. Some new sharp inequalities characterizing parallelotopes in R~n are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in R~n is provided.