A note on the expression of modulus of convexity

Abstract

‎It is well known that different kinds of expressions of modulus of‎ ‎convexity are essentially based on two geometrical propositions‎. ‎For‎ ‎one of the propositions we first present a new proof by the‎ ‎Hahn-Banach theorem and intermediate value theorem‎, ‎then give some‎ ‎corollaries to it which lead to some new expressions of modulus of‎ ‎convexity‎.