Determination of convex bodies by projection functions
The author proves some interesting results related to projection functions of convex bodies, i.e., related to the question to what extent a compact, convex set with interior points in \(\mathbb{R}^d\), \(d\geq 2\), is (almost) determined by its \(i\)th projection function defined on the Grassmannian of \(i\)-dimensional linear subspaces of \(\mathbb{R}^d\), with \(i\in \{1, \dots, d-1\}\). E.g., it is shown that a convex body \(K\) is almost determined by its brightness function (i.e., \(i=d-1)\) if and only if \(K\) is a polytope with \(f_{d-1} (K)<2d+2\) and \(f_{d-1} (\nabla K)\leq 2d+2\), where \(f_{d-1}\) denotes the number of facets and \(\nabla K\) is written for the Blaschke body of \(K\). (If the Blaschke sum \((1-\alpha)^{1/d-1} K\#\alpha^{1/d-1}K^*\), \(0\leq \alpha\leq 1\), of \(K\) and its reflection \(K^*\) at the origin as well as translates of it are the only convex bodies having the same brightness function as \(K\), then \(K\) is said to be almost determined by this function.) Further on, it is proved that a convex \(d\)-polytope, \(d\geq 4\), is determined by its \(i\)-th projection function, \(i\in\{2, \dots, d-2\}\), if all its \((i+1)\)-faces are parallelepipeds.
scientific article; zbMATH DE number 1241004