Generalized rectangular constant in Banach spaces

This paper presents two new geometric constants \(\mu(X,a)\) and \(\mu'(X,a)\), which extend the rectangular constants \(\mu(X)\) and \(\mu'(X)\). We firstly provide their bounds. Then the relationships between these geometric constants and the geometric properties of Banach spaces are discussed, including uniform nonsquareness, uniform convexity and uniform smoothness. Meanwhile, we provide several estimates of \(\mu(l_p,a)\) and obtain some new upper bound estimates on \(\mu'(l_p,a)\).