Affine diameters of convex bodies (Q2789877)

The authors are concerned with the study of affine diameters through the points of a convex body \(K\) in \({\mathbb R}^n\). A chord of an \(n\)-dimensional convex body \(K\) in \({\mathbb R}^n\), that is, a one-dimensional intersection of \(K\) with a line, is called an affine diameter of \(K\) if its endpoints lie in distinct parallel supporting hyperplanes of the body. For a point \(z\in K\), \(N_a(K,z)\) denotes the number of affine diameters passing through \(z\) and NEWLINE\[NEWLINE N_a(K)=\frac{1}{V_n(K)}\int_{K} N_a(K,z)dz NEWLINE\]NEWLINE is the average number of affine diameters through the points of a convex body \(K\).NEWLINENEWLINEThe authors establish the following two results.NEWLINENEWLINE{ Theorem 1. } Let \(P\subset {\mathbb R}^n\) be an \(n\)-polytope such that \(P\) and \(-P\) are in \((n-1)\)-general relative position. Then NEWLINE\[NEWLINE N_a(P)=\frac{n+1}{V_n(P)}\int_0^1 V_n((1-t)P-tP) dt-1. NEWLINE\]NEWLINE In particular, NEWLINE\[NEWLINE n<N_a(P)\leq 2^n-1. NEWLINE\]NEWLINE Equality on the right-hand side holds if and only if \(P\) is a simplex. The lower bound \(n\) is sharp, but is not attained.NEWLINENEWLINE{ Theorem 2. } Let \(K\subset {\mathbb R}^2\) be a two-dimensional convex body such that \(K\) and \(-K\) are in general relative position. Then NEWLINE\[NEWLINE 1\leq N_a(K)\leq \frac{V_2(K-K)}{2V_2(K)}\leq 3. NEWLINE\]NEWLINE Equality on the left side is attained if and only if \(K\) is centrally symmetric. Equality on the right side is attained if and only if \(K\) is a triangle.