On convex bodies in , , with directly congruent projections

Abstract

Let $$ and let $$ and $$ be two convex bodies in $$ such that their orthogonal projections $$ and $$ onto any $$-dimensional subspace $$ are directly congruent, that is, there exists a rotation $$ and a vector $$ such that $$. Assume also that the 2-dimensional projections of $$ and $$ are pairwise different and they do not have $$-symmetries. Then $$ and $$ are congruent. We also prove an analogous more general result about twice differentiable functions on the unit sphere in $$.