Extensions of an inequality of Bonnesen to d-dimensional space and curvature conditions for convex bodies (Q580720)

For K a convex body in Euclidean space \(E^ d\) (d\(\geq 3)\), let \(W_ i\) be its quermassintegrals and D its diameter. The main result says that the greatest possible constant \(\beta_ d\) such that \(W_{d-2}- \beta_ d(D/2)W_{d-1}+ (\beta_ d-1)(D^ 2/4)W_ d\leq 0\) for all K is greater than 1. The equality occurs only for balls and for bodies which are similar to a certain body of revolution with constant sum of main curvature radii. For related papers see \textit{J. Bokowski} and the author [Arch. Math. 47, 79-89 (1978; Zbl 0576.52005)] and \textit{J. R. Sangwine-Yager} [``Bonnesen-style inequalities for Minkowski relative geometry'', to appear elsewhere].