Sampling convex bodies: a random matrix approach

We prove the following result: for any , only sample points are enough to obtain -approximation of the inertia ellipsoid of an unconditional convex body in . Moreover, for any , already sample points give isomorphic approximation of the inertia ellipsoid. The proofs rely on an adaptation of the moments method from Random Matrix Theory.

     

Inequalities for convex bodies and their polar bodies

The infimum of the quermassintegral product W _i (K)W _i (K*) for i = n – 1 was established by Lutwak. In this paper, the infimum of the dual quermassintegral product ${\widetilde{W}_{n+p}(K)\widetilde{W}_{n+p}(K^*)}$ for any p ≥ 1 is obtained, and some new inequalities about convex bodies and their polar bodies are established.     

Isoperimetric problems for convex bodies and a localization lemma

We study the smallest number ψ(K) such that a given convex bodyK in ℝ ⁿ can be cut into two partsK ₁ andK ₂ by a surface with an (n−1)-dimensional measure ψ(K) vol(K ₁)·vol(K ₂)/vol(K). LetM ₁(K) be the average distance of a point ofK from its center of gravity. We prove for the “isoperimetric coefficient” that

A width-diameter inequality for convex bodies

A special case of the Blaschke-Santaló inequality regarding the product of the volumes of polar reciprocal convex bodies is shown to be equivalent to a power-mean inequality involving the diameters and widths of a convex body. This power-mean inequality leads to strengthened versions of various known inequalities.     

Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Orientation-dependent section distributions for convex bodies

In the paper the orientation dependent chord length distribution functions for some bounded convex domains are obtained. In particular, formulae for the orientation dependent chord length distributions for a regular polygon and an ellipse are obtained. Explicit forms of the orientation dependent chord length distributions were known only in the cases of a disc and a triangle. We also obtain the cross-section area distribution functions for an ellipsoid and a cylinder. The cross-section area distribution function was known only in the case of a ball.     

Zonal modeling of radiative heat transfer in industrial furnaces using simplified model for exchange area calculation

The zonal analysis of industrial furnaces is considered with three-dimensional radiative heat transfer, incorporated with the mathematical zone method. In this method exchange areas are determined by simplified numerical integration in three dimensions for surface-surface, surface-gas and gas–gas zones for absorbing and emitting media. By focusing on new strategies to overcome the drawbacks of evaluating direct exchange areas, it is shown that the zone method is an effective numerical method for modeling three-dimensional thermal performance of gas-filled enclosures. Also the developed method for evaluating of exchange area is presented and compared with other methods in both sides of CPU time and accuracy. The method can decrease about 70% in error of calculation of some exchange areas as compared with the other numerical methods.     

Sectional bodies associated with a convex body

We define the sectional bodies associated to a convex body in and two related measures of symmetry. These definitions extend those of Grünbaum (1963). As Grünbaum conjectured, we prove that the simplices are the most dissymmetrical convex bodies with respect to these measures. In the case when the convex body has a sufficiently smooth boundary, we investigate some limit behaviours of the volume of the sectional bodies.

     

A sharp isoperimetric bound for convex bodies

We consider the problem of lower bounding the Minkowski content of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp. Supported in part by VIGRE grants at Yale University and the Georgia Institute of Technology.     

The central limit problem for convex bodies

It is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately Gaussian. Several quite broad classes of bodies are shown to satisfy the condition.

     

Pointwise estimates for marginals of convex bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E⊂Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c>0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.     

The illumination conjecture for spindle convex bodies

The main principle of affine quantum gravity is the strict positivity of the matrix _boxclose_ab (x) \{ \hat g_{ab} (x)\} composed of the spatial components of the local metric operator. Canonical commutation relations are incompatible with this principle, and they can be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational constraint operators is formulated quite naturally as a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models.     

Invariant convex bodies for strongly elliptic systems

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems, no a priori restrictions on the coefficient matrices are imposed.     

Two inequalities for volumes of convex bodies

Precise upper and lower bounds are given for that portion of a convex body cut off from the latter, in n-dimensional space, by a hyperplane passing through its centroid; a bound is also given for the whole volume in terms of relative n-diameters of Bernstein type.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 99–106, January, 1969.