The average number of normals through a point in a convex body and a related Euler-type identity
A normal to a convex body \(K\) at a point \(x\) in the boundary of \(K\) is a ray with end point at \(x\), perpendicular to a support plane \(H\) of \(K\) at \(x\), and contained in the halfspace bounded by \(H\) that contains \(K\). For each point \(p\) in \(K\) let \(n(K,p)\) be the number of normals to \(K\) passing through \(p\). Then the average number of normals through a point in \(K\) is defined by \[ n(K)={I(K)\over V(K)} \quad \text{where} \quad I(K)=\int_ K n(K,p)dV, \] \(V(K)\) denoting the volume of \(K\). The problem to find bounds for \(n(K)\) has been solved completely by \textit{G. D. Chakerian} [`The number of diameters through a point inside an oval', Unión Matemática Argentina 29, 282-290 (1984)] by showing \(2\leq n(K)\leq 2\pi/(\pi-\sqrt 3)\) with equality holding in the upper bound only for the Reuleaux triangle and in the lower bound only for a circular disk. In this paper bounds for more general cases are obtained. In the planar case the author shows for convex domains being bounded by a sufficiently smooth convex curve or by a polygon that \(n(K)\leq 12\) (in general), \(n(K)\leq 8\) (centrally symmetric), \(n(K)\leq 6\) (centers of curvature inside K). We have \(n(K)=8\) for regular \(2n\)-gons. In higher dimensions \(m\) she proves \(n(K)\leq{V(K+DK)\over V(K)}-1\) (sufficiently smooth or polytope) and \(n(K)\leq 3^ m-1\) (centrally symmetric in addition) where \(DK\) denotes the difference body of \(K\); \(n(K)=3^ m-1\) for the \(m\)- cube. Finally an Euler type relation is studied in general which emanated from the proof of the bound for polygons.
scientific article; zbMATH DE number 486898
scientific article