Characterizations via linear combinations of curvature measures (Q1383697)

Characterization theorems for the euclidean sphere still represent an interesting and developing branch of differential geometry. In this paper the author discusses the measure theoretical extensions of well known results for ovaloids. Such a smoothly bounded convex body must be a ball if a single normalized mean curvature \(H_r\) is constant. Schneider proved the generalized version of this result: A compact convex body with one curvature measure being a constant multiple of the boundary volume measure is a ball [\textit{R. Schneider}, Comment. Math. Helv. 54, 42-60 (1979; Zbl 0392.52004)]. For ovaloids with \(\sum a_rH_r \equiv c > 0\) and \(a_r > 0\) the sphericity of the boundary has been proved by K. Voss with different methods assuming differentiability properties. The author proves the measure theoretical analogue for this result, also giving a simple proof of Schneider's characterization theorem. The proof relies on generalized Minkowski integral formulas, a volume representation via the interior reach function, and the isoperimetric inequality.