Geometric Constant Of Banach Space

The geometric constants of Banach space are very important tool forresearching the properties of Banach space. It’s more accurate to describe Banachspace of geometric structure by geometric constants. Whether specific Banachspaces have geometric properties depend on the values of geometric constants ofthis Banach space or not. Many scholars discuss the calculations and estimationsof some geometric constants. In this paper, supposing X is a real Banach space,we discuss the Zbagaun constant C Z(X)that was introduced for researchinginner product space by Zbagaun in2002.This paper main researched problem of values of geometric constants ofBanach spaces. The main results of this paper are summarized as following:First of all, we introduce definitions of some geometric constants, therelationship between some geometric constants and some geometric properties.We obtain when Zbagaun constant and non-square constant is less than two thesituation is same.Second, we introduce Orlicz space and we estimate the value of Zbagaunconstant of Orlicz space. We use this result to estimate the Zbagaun constant ofL pspace.In the end, we estimate the value of two-dimensonal Day-Jamesspace(l fl1). We attain a upper bounded and prove that the value is just the exactvalue of Zbagaun constant in the two-dimensonal Day-James space(l fl1).