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We will introduce a new geometric constant GL(X) based on the constant H(X)proposed by Gao. We first further survey the constant H(X) and discuss some ofthe properties of this constant that have not yet been discovered. Next, wefocus on a new constant GL(X) along with some of its basic properties. Inaddition, we show some relations between the well-known geometric constants andGL(X) through some inequalities. Finally, we characterize some generalizedforms of the constant GL(X).

Inscribed triangles in the unit sphere and a new class of geometric constants