Inequalities for mean circumscribing simplices (Q1362563)

For a convex body \(K\subset \mathbb{R}^n\) and a rotation \(g\in SO(n)\), let \(gK= \{gx:x \in K\}\) denote the image of \(K\) under \(g\), and for \(S\) an \(n\)-simplex, \(S(gK)\) is written for the \(n\)-simplex circumscribed about \(gK\) and having the same outer normals as \(S\). For the classical quermassintegrals \(W_i(K)\) of \(K\), J. Sangwine-Yager established the so-called circumscribed simplex inequality \[ \int_{SO(n)} W_{n-i} \bigl(S(gK)\bigr) dg\geq W_{n-i} (S)W_{n-i} (K)/ \omega_n \] with equality for \(1<i\leq n\) if and only if \(K\) is a ball (for \(i= 1\) both sides are equal for all \(K)\). The author gives generalizations of this inequality, also in view of a much larger class of functionals (than quermassintegrals), namely the class of nonnegative, translation invariant functionals \(\Phi_i\) that are homogeneous (of a fixed real degree \(i>1)\). E.g., the inequality \[ \int_{SO(n)} \Phi_i\bigl(S(gK) \bigr)dg \geq\Phi_i(S) [W_{n-1} (K)/ \omega_n]^i \] is proved, where equality holds if and only if either \(K\) is a ball, or \(S\) is a regular simplex and some homothet of \(K\) is a rotor of \(S\).