Convex hypersurfaces with transnormal horizons are spheres

Abstract

Let M be a smooth $(=C^∈fty)$, compact, connected hypersurface of Euclidean $(n+1)$-space $Rⁿ⁺¹$,$n≥ 2$, with nowhere-zero Gaussian curvature. Thus M is differeomorphic to the n-sphere $Sⁿ$ and every affine tangent hyperplane meets M in just one point.Let λ be any (straight) line in $Rⁿ⁺¹$ and let $M_λ$ denote the set of points of M at which the tangent hyperplane is parallel to λ.We call $M_λ$ the λ-horizon of M. If, for every λ, $M_λ$ is a transnormal submanifold of $Rⁿ⁺¹$ [5] we shall say that M is horizon-transnormal.In this paper we show that if M is horizon-transnormal then M is a round sphere.The converse is obviously true.We show in §2 that if M is horizon-transnormal then it is transnormal.If M is transnormal then every λ-outline ω_λ$ (see §1 below) is also transnormal.This is one aspect of a classical result in the theory of convex bodies of constant width [1] but we give a direct differential-geometric proof in §3. We prove in §4 that each λ-horizon $M_λ$ is contained in a hyperplane normal to λ.It is then a consequence of a classical result that M must be an n-ellipsoid. Consequently, due to its transnormality, M is a round n-sphere.