Stationarity equations for fractional affine area measures: ellipsoid rigidity and polytopal solutions

Abstract

Xi and Zhao established a first variation formula for affine inequalities involving radial mean bodies, and asked two geometric questions about the resulting stationarity equation: whether the bodies already known to satisfy it are the only ones, and whether its smooth solutions are exactly the ellipsoids. We answer both. Every strictly convex solution of the equation with $q>-1$, $q\ne0,n$ is an ellipsoid, which answers the second question affirmatively while replacing smoothness and everywhere positive Gauss curvature by strict convexity. Every full-dimensional facet-transitive polytope satisfies the equation, and so does the polar of every vertex-transitive polytope with respect to its volume centroid; the rhombic dodecahedron, the thirteen Catalan solids and the bipyramids over regular polygons other than the octahedron provide infinitely many affinely inequivalent solutions that are not affine images of regular polytopes or balls, so the answer to the first question is negative. The rigidity proof compares overlap integrals along a continuous Steiner path: stationarity forces an increase of order $o(t)$, whereas a nonaffine chord midpoint function forces an increase of order $t$. For negative parameters the required comparison rests on a minimization formula over bounded measurable auxiliary sets.