Bartłomiej Zawalski
Case Western Reserve University
Title
ON STAR-CONVEX BODIES WITH ROTATIONALLY INVARIANT SECTIONS
Abstract
In 1933, Banach conjectured that a symmetric convex body whose central sections are pairwise linearly equivalent must be an ellipsoid. Work by Auerbach, Mazur, and Ulam, and later by Gromov, revealed that the most challenging case is when all central sections possess non-trivial symmetries. In this talk, we will outline a proof that an origin-symmetric star-convex body K with sufficiently smooth boundary is a body of affine revolution, provided that every hyperplane section of K passing through the origin is itself a body of affine revolution. This result provides a positive answer, under an additional smoothness assumption, to a question posed by G. Bor, L. Hern´andez-Lamoneda, V. Jim´enez de Santiago, and L. Montejano-Peimbert in their work on the isometric Banach conjecture. Our theorem can also be viewed as a high-dimensional analogue of Bezdek’s conjecture. The proof relies primarily on tools from differential geometry and linear algebra, with occasional use of more sophisticated techniques from other areas, including algebraic topology and commutative algebra
