arXiv:1908.05952v3 [math.DG] 17 Mar 2020 The Heintze-Karcher-Ros inequality and the soap bubble theorem in geometric measure and convex geometry Mario Santilli March 18, 2020 Abstract We prove an isoperimetric-type inequality (the Heintze-Karcher-Ros inequality) for general convex bodies and for a class of mean convex sets defined in a viscosity sense. The latter includes the mean convex sets introduced by White in the regularity theory of the level set flow. In both cases we obtain the soap bubble theorem analyzing the equality case. 1 Introduction Background One of the most celebrated result in differential geometry is the Alexandrov soap bubble theorem, which asserts that a compact, embedded, smooth hyper- sursurface in R n+1 is a sphere, provided it has constant scalar mean curvature; cf. [Ale58]. Thirty years later, Ros in [Ros87], Korevaar-Ros in [Ros88] and Montiel-Ros in [MR91] generalize Alexandrov theorem, proving that a compact and embedded smooth hypersurface is a sphere, provided that one of the higher- order mean curvatures is constant. All these soap-bubble theorems turns out ot be special cases of an isoperimetric-type result, which asserts that if Ω is a bounded open set with smooth boundary and ∂ Ω has non-negative scalar mean curvature h (i.e. Ω is a mean convex set), then (1) L n+1 (Ω) ≤ n n +1  ∂Ω 1 h dH n , with equality if and only if Ω is a sphere. This result is proved in [Ros87, Theorem 1] and in [MR91], using two different methods. Similar inequalities were previously obtained by Heintze and Karcher in [HK78]. We refer to the inequality as the Heintze-Karcher-Ros inequality and to the characterization of the equality case as Ros soap bubble theorem (to distinguish it from its special case due to Alexandrov). Motivated by questions in general relativity, Bren- dle has recently obtained an important extension of these results to a class of warped-product spaces; cf. [Bre13]. 1