In this paper, we firstly investigate the constant H(X) proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant GL(X) closely related to H(X), along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms GL(X,p) and CL(X) explicitly.