Simplices with edges of equal length in finite dimensional Banach spaces (Q1919318)

Let \(E\) be the unit ball of a \(d\)-dimensional Minkowski space, and suppose \(S\) is a simplex in that space with all edges of (Minkowskian) length one. The authors are interested in estimates for the quotient \(V_d(S)/V_d (E)\), where \(V_d\) is an affine volume. For \(d\geq 2\) they prove that \(V_d(S)/V_d (E)\leq {2d \choose d}^{-1}\), with equality if and only if \(E\) is the difference body of \(S\). For \(d\geq 3\) they show that \(V_d(S)/V_d(E)\) can be arbitrarily small (for \(d=2\), this quotient is \(\geq 1/8)\). If \(d\geq 3\) and \(P\) is a centered parallelotope, the quotient \(V_d(S)/V_d(P)\) can attain every value in the interval \((0,(d-1)/d! 2^d)\).