Stability theorems for projections and central symmetrization (Q810876)

Let K, L be convex bodies in n-dimensional Euclidean space (n\(\geq 3)\), and let \(h_ t(K,L)\) denote their translative Hausdorff distance. As a consequence of more general results, the author obtains the stability estimate \[ h_ t(K,L)^{r+n-1}\leq \delta_{n,r}D_ 0^{n- 1}(1/\sigma_ n)\int_{S^{n-1}}h_ t(\pi_ uK,\pi_ uL)^ rd\sigma (u) \] for any \(r>0\). Here \(\delta_{n,r}\) is an explicitly given constant, \(D_ 0\) is the maximum of the diameters of K and L, \(\pi_ u\) denotes orthogonal projection onto the hyperplane through 0 orthogonal to \(u\in S^{n-1}\), the unit sphere, \(\sigma\) is spherical Lebesgue measure on \(S^{n-1}\), and \(\sigma_ n=\sigma (S^{n-1})\). From this result, the author deduces a stability result for the behaviour of the quermassintegrals under central symmetrization.