On perimeters of sections of convex polytopes (Q990828)

Let \(P\) and \(Q\) be convex polytopes in \({\mathbb R}^n\) (i.e., convex hulls of finitely many points in \({\mathbb R}^n\)) that are symmetric with respect to the origin, where \(n \geq 3\). For a \(k\)-dimensional polytope \(B\), let \(S(B)\) denote the \((k-1)\)-dimensional volume of the boundary of \(B\). The main result of the paper under review states that if \[ S(P \cap H) = S(Q \cap H) \] for every \(k\)-dimensional subspace \(H\) of \({\mathbb R}^n\), \(2 \leq k \leq n-1\), then \(P = Q\). The key idea in the proof is that \(S(P \cap H)\), considered as a function of a hyperplane \(H\), detects vertices of \(P\), in the sense that this function enjoys certain analytic properties that break once \(H\) hits a vertex. The motivation of this theorem arises from a problem in [\textit{R. J. Gardner}, Geometric Tomography. Encyclopedia of Mathematics and Its Applications. 58. Cambridge: Cambridge University Press. (1995; Zbl 0864.52001)], which concerns the special case \(n=3\) where, however, \(P\) and \(Q\) are allowed to be arbitrary origin-symmetric convex bodies. This general problem remains open even if \(P\) is the Euclidean ball.