An inequality for simplices (Q1076346)

Let \(U=(u_ 0,u_ 1,...,u_ m)\) be an m-simplex, i.e. an \((m+1)\)-tuple of (not necessarily distinct) points in \({\mathbb{R}}^ n\). The body [U] of U is the set of all points x of \({\mathbb{R}}^ n\) that are convex combinations of \(u_ 0,u_ 1,...,u_ m\). An r-face of U is any simplex \((u_{i_ 0},...,u_{i_ r})\) where \(i_ 0,...,i_ m\) are integers with \(0\leq i_ o<...<i_ r\leq m.\) The author proves the following Theorem. Suppose \(1\leq r\leq m\). Let S, T be m-simplices in \({\mathbb{R}}^ n\), and suppose [S]\(\subset [T]\). Let \(B_{m,r}=q^{r+1-s}(q+1)^ s/(m+1-r),\) where q and s are the quotient and remainder on division of \(m+1\) by \(r+1\); i.e. where q,s are integers such that \(m+1=q(r+1)+s\) and such that \(0\leq s<r+1.\) Let \(\mu_ s(S)\) be the total r-dimensional measure of all r- dimensional faces of S (similarly we define \(\mu_ r(T))\). Then (1) \(\mu_ r(S)=\leq B_{m,r}\mu_ r(T).\) Furthermore, for any \(r\leq n\) there exist m-simplices S, T for which equality is attained in (1); therefore the bound \(B_{m,r}\) is best possible. This result is a generalization of a result of \textit{W. Holsztyński} and \textit{W. Kuperberg} [see: Wiadom. Mat. 6, 14-16 (1962; Zbl 0126.369) and Alabama J. Math. 1, 40-42 (1977)].