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 Rn are established.
Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes
in Rn is provided.