Abstract:

Convex geometry is an important branch of modern geometry. Convex bodies and star bodies are a main object of study with many unsolved open problems and conjectures. Although these problems and conjectures are different, they are actually equivalent. In this paper, we discuss the connection of several conjectures such as isotropic constant conjecture, slicing problem, Busemann-Petty problem, and reverse Brunn-Minkowski inequality. We give proofs of the equivalence.

Key words: convex body; isotropic body; isotropic constant; hyperplane section