A characterization of parallel ovaloids

Abstract:

Two ovaloids $S,\bar S$ can be mapped diffeomorphically onto each other by equal inner normals. If, under this mapping, principal directions are preserved and \[ [(p - \bar p) - (ck_1^{ - 1} - \bar k_1^{ - 1})][(p - \bar p) - (ck_2^{ - 1} - \bar k_2^{ - 1})] \leq 0\] everywhere on the unit sphere for a certain constant $c$, then $c = 1$ and $p - \bar p$ = constant. Here $p,\bar p$ are the support functions, ${k_1}$, and ${k_2}$ the principal curvatures of $S,{\bar k_1}$ and ${\bar k_2}$ the corresponding principal curvatures of $\bar S$. Various characterizations of the sphere are obtained as corollaries.