Convex sets with large distortion (Q1021361)

Let \(C\in\partial\mathcal L\) be a boundary point of a compact convex body \(\mathcal L\) in an \(m\)-dimensional Euclidean vector space with a fixed base point \(\mathcal O\) in int\(\mathcal L\). With the line \(l({\mathcal O}C)\) and \(C^o:=l({\mathcal O}C)\cap \partial{\mathcal L}\setminus \{C\}\) the distortion function \(\Lambda\) is defined by \(\Lambda(C):= d(C,{\mathcal O}) / d(C^o,\mathcal O)\)where \(d\) denotes the Euclidean distance function. For appropriate \(\mathcal O\) this function satisfies \(1/m \leq \Lambda \leq m\). With the aid of \(\Lambda\) an affine invariant \(\sigma (\mathcal L)\) can be defined that measures how distorted \(\mathcal L\) is with respect to \(\mathcal O\). The author continues his earlier investigations of convex bodies [see, for example, Trans. Am. Math. Soc. 358, No. 6, 2425--2446 (2006; Zbl 1108.53032)]. In the present paper he proves that if \(\sigma ({\mathcal L}) < 1+1/(m+1)\) then \(\mathcal L\) is between two special homothetic simplices. That gives an upper bound on the volume of \(\mathcal L\) in terms of \(\sigma(\mathcal L)\) and the inscribed simplex.