2016/11/29 by Myroshnychenko, Sergii, Ryabogin, Dmitry, Saroglou, Christos · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1611.09443
We say that a star body K is completely symmetric if it has centroid at the origin and its symmetry group G forces any ellipsoid whose symmetry group contains G, to be a ball. In this short note, we prove that if all central sections of a star body L are completely symmetric, then L has to be a ball. A special case of our result states that if all sections of L are origin symmetric and 1-symmetric, then L has to be a Euclidean ball. This answers a question from \citeR2. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function f defined on the sphere, are isotropic functions, then f is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \citeR and \citeR2 are answered.