On the structure of convex sets with symmetries
The author studies a measure of symmetry \(\sigma\) associated to an \(n\)-dimensional convex body \(\mathcal L\) and a point \(\mathcal O\) in the interior of \(\mathcal L\). This measure is introduced as follows. If \(C\) is a boundary point of \(\mathcal L\), let \(C^0\) be the boundary point of \(\mathcal L\) distinct from \(\mathcal O\) and such that \(C, \mathcal O\) and \(C^0\) are collinear. By \(\Lambda(C,\mathcal O)\) we denote the ratio of the distance between \(C\) and \(\mathcal O\) to the distance between \(C^0\) and \(\mathcal O\). Now consider the convex hull \(P\) of points \(C_0,\ldots,C_n\) on the boundary of \(\mathcal L\) such that \(\mathcal O \in P\). Then \(\sigma(\mathcal L,\mathcal O)\) is defined as the infimum of (\({c_{i} val}\)) \(\sum_{i=0}^n \frac{1}{1+\Lambda(C_{i},\mathcal O)}\) over all \(C_0,\ldots,C_n\) as above. It can be shown that \(1 \leq \sigma(\mathcal L,\mathcal O) \leq \frac{n+1}{2}\) with the equality on the left attained if and only if \(\mathcal L\) is a simplex and inequality on the right attained if and only if \(\mathcal L\) is centrally symmetric with respect to \(\mathcal O\). The definition of \(\sigma\) allows to split the interior of the convex body \(\mathcal L\) into two classes of points. The point \(\mathcal O\) is said to be a \textit{regular point} of \(\mathcal L\) if equality of (ref \({c_i val}\)) and \(\sigma(\mathcal L,\mathcal O)\) implies that \(P\) (defined above) is a simplex and \(\mathcal O\) is in the interior of \(P\). Otherwise, \(\mathcal O\) is called a \textit{singular point} of \(\mathcal L\). The author shows that simplices (i.e., bodies \(\mathcal L\) with \(\sigma(\mathcal L,\mathcal O)=1\)) have interior consisting of regular points only and centrally symmetric convex bodies (i.e., bodies \(\mathcal L\) with \(\sigma(\mathcal L,\mathcal O)=(n+1)/2\)) have interior consisting of singular points only. (This result follows from a more general theorem by the author.) The author conjectures that the converse should also be true, i.e., a convex body with all interior points regular is necessarily a simplex; a convex body with all interior points singular is necessarily centrally symmetric. It is known that for \(n=2\) the function \(\sigma({\mathcal L}, . )\) is concave on the interior of \(\mathcal L\). The author proved a result which implies that for every \(n\geq 3\) and certain \(\mathcal L\) the function \(\sigma({\mathcal L},\, .\,)\) is not concave on the interior of \(\mathcal L\) (for example when \(\mathcal L\) is a cube).
scientific article; zbMATH DE number 5664573