On the Reflexive Polytope and its Application

The reflexive polytope in Rd is a very important subject in different areas of mathematics. Chapter one some fundamental concepts about the polyhedron, polytope, reflexive polytope, Ehrhart theory, and some definitions and examples about the group are given. In chapter two, the h-vector and reflexive polytope in view of combinatorics with detailed explanation of them, and the relationship between them, are studied, and a new proposition for the bound of h-vectors is obtained. Chapter three, the new concepts with some theorems that used them as techniques to prove our conjecture about the maximum number of vertices of reflexive polytope is given. To the best our knowledge, this result seems to be new. The analysis should be useful to professionals in combinatorial, algebraic geometry, cryptography, and integer programming.