The isotropy constant and boundary properties of convex bodies (original) (raw)

Some isoperimetric comparison theorems for convex bodies in Riemannian

Indiana University Mathematics Journal, 2005

We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (M n+1 , g) satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Ω. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality we obtain sharp comparison theorems: not only can we derive an inequality which should be compared with Lévy-Gromov Inequality but we also show that if Ric nδ on Ω, then the profile of Ω is bounded from above by the profile of the half-space H n+1 δ in the simply connected space form with constant sectional curvature δ. As consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of Ω, and for the first non-zero Neumann eigenvalue for the Laplace operator on Ω.

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

Geometriae Dedicata, 1996

For a convex body K⊂ℝd we investigate three associated bodies, its intersection body IK (for 0∈int K), cross-section body CK, and projection body IIK, which satisfy IK⊂CK⊂IIK. Conversely we prove CK⊂const1(d)I(K−x) for some x∈int K, and IIK⊂const2 (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L⊂ℝd a convex body, we take n random segments in L, and consider their ‘Minkowski average’ D. We prove that, for V(L) fixed, the supremum of V(D) (with also n∈N arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M⊂ℝd a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1≤pV(L) fixed, by the ellipsoids. For k=2, the supremum (n∈N arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.

Anisotropic curvature measures and uniqueness of convex bodies

2021

We prove that an arbitrary convex body C ⊆ R, whose k-th anisotropic curvature measure (for k = 0, . . . , n− 1) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated Wulff shape. This result provides a generalization of a theorem of Schneider (1979) and resolves a conjecture of Andrews and Wei (2017).

Some sharp isoperimetric theorems for Riemannian manifolds

Indiana University Mathematics Journal, 2000

We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using differential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.

A problem of Klee on inner section functions of convex bodies

Journal of Differential Geometry, 2012

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by con- structing two infinitely smooth convex bodies

Quantitative isoperimetric inequalities for a class of nonconvex sets

Calculus of Variations and Partial Differential Equations, 2010

Quantitative versions (i.e., taking into account a suitable "distance" of a set from being a sphere) of the isoperimetric inequality are obtained, in the spirit of , for a class of not necessarily convex sets called ϕ-convex sets. Our work is based on geometrical results on ϕ-convex sets, obtained using methods of both nonsmooth analysis and geometric measure theory. ] deal with sharp estimates of the correction term a. In [17], a is given using the spherical deviation d(K) of K, which is actually the Hausdorff distance between a suitable normalization of K and the unit ball. The result in [17] is very precise, but requires strong assumptions on K. In particular, K needs to be "nearly spherical", in the sense that both its Hausdorff distance from the unit ball and the norm of the gradient of a suitable representation of the boundary must be small. The result in [17] applies mainly to the class of compact convex sets K with nonempty interior having "isoperimetric deficiency" ∆(K) (see Definition 2.2 below) small enough. The paper , instead, deals with general Borel sets with finite n-dimensional measure and finite perimeter. Of course the term a needs to be modified, as there is no geometric assumption on K. Yet, the authors succeed to give a sharp estimate of a in terms of the so called Fraenkel asymmetry of K (see Definition 2.4 below), which is essentially the Lebesgue measure of the symmetric difference between K and a suitable sphere. This solves a conjecture appearing in , which was the first paper considering quantitative isoperimetric inequalities for general sets in R n .