Polytopal approximation of elongated convex bodies

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex bodyKby a circumscribed polytopePwith a given number of facets. These bounds are of particular interest ifKis elongated. To measure the elongation of the convex set, its isoperimetric ratioVj(K)1/jVi(K)−1/iis used.

On submanifolds of highly negatively curved spaces

We prove spectral, stochastic and mean curvature estimates for complete m-submanifolds φ : M → N of n-manifolds with a pole N in terms of the comparison isoperimetric ratio Im and the extrinsic radius rφ ≤ ∞. Our proof holds for the bounded case rφ < ∞, recovering the known results, as well as for the unbounded case rφ = ∞. In both cases, the fundamental ingredient in these estimates is the integrability over (0, rφ) of the inverse [Formula: see text] of the comparison isoperimetric radius. When rφ = ∞, this condition is guaranteed if N is highly negatively curved.