Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\epsilon)$ be the spherical cap with height $\epsilon$ I am interested in the quantity $$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in }B_n(\epsilon)}\mathrm{vol}(\Delta)$$ I think this should be a classical result and have very nice upper and lower bounds on the order $\epsilon^{(n+1)/2}$. The thing I am interested in knowing some good bounds for the constants in terms of $n$, but unfortunately I wasn't able to settle down with a good reference for this. Any help would be greatly appreciated.

Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\varepsilon)$ be the spherical cap with height $\varepsilon$ I am interested in the quantity $$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in }B_n(\varepsilon)}\mathrm{vol}(\Delta)$$ I think this should be a classical result and have very nice upper and lower bounds on the order $\varepsilon^{(n+1)/2}$. The thing I am interested in knowing some good bounds for the constants in terms of $n$, but unfortunately I wasn't able to settle down with a good reference for this. Any help would be greatly appreciated.