Error of Asymptotic Formulae for Volume Approximation of Convex Bodies in

 We estimate the error of asymptotic formulae for volume approximation of sufficiently differentiable convex bodies by circumscribed convex polytopes as the number of facets tends to infinity. Similar estimates hold for approximation with inscribed and general polytopes and for vertices instead of facets. Our result is then applied to estimate the minimum isoperimetric quotient of convex polytopes as the number of facets tends to infinity.     

Error of Asymptotic Formulae for Volume Approximation of Convex Bodies in

 We estimate the error of asymptotic formulae for volume approximation of sufficiently differentiable convex bodies by circumscribed convex polytopes as the number of facets tends to infinity. Similar estimates hold for approximation with inscribed and general polytopes and for vertices instead of facets. Our result is then applied to estimate the minimum isoperimetric quotient of convex polytopes as the number of facets tends to infinity. Received 16 July 2001     

Geometric Measures for Random Curved Mosaics of Rd

This paper deals with stationary random mosaics of R^d with general cell shapes. As geometric measures concentrated on the i-skeleton (i = 0, 1,…,d) the i-dimensional surface area (volume) measure and (i — 1) different curvature measures are chosen. The corresponding densities are calculated as well as for the mosaics and their superpositions in terms of mean cell parameters and mean cell numbers. This leads to various relations between the characteristic which are applied, in particular, to two- and three-dimensional tessellations. A comparison with known formulas for mosaics with convex cells in R² and R³ is given.     

Inequalities for mixed p-affine surface area

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of L _p affine surface areas, mixed p-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.     

从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov, Ledoux [S.G. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (5) (2000) 1028-1052] and the isoperimetric inequalities due to Bakry, Ledoux [D. Bakry, M. Ledoux, Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (2) (1996) 259-281] and Bobkov, Zegarliński [S.G. Bobkov, B. Zegarliński, Entropy bounds and isoperimetry, Mem. Amer. Math. Soc. 176 (829) (2005), x+69]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov, Milman [M. Gromov, V.D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compos. Math. 62 (3) (1987) 263-282].     

Volumes of symmetric random polytopes

We consider the moments of the volume of the symmetric convex hull of independent random points in an n-dimensional symmetric convex body. We calculate explicitly the second and fourth moments for n points when the given body is (and all of the moments for the case q = 2), and derive from these the asymptotic behavior, as , of the expected volume of a random simplex in those bodies. Received: 5 February 2003     

Extremal Polygons with Minimal Perimeter

In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variations of the isoperimetric problem for this class of polyhedra. The first one is: Which bodies have minimal surface area in the class of extremal bodies for a fixed n-dimensional lattice? And the second one is: Which bodies have minimal surface area in the class of extremal bodies with volume 1 of dimension n? We characterize the solutions of these two problems in the plane. There is a consequence of these results, the solutions of the above problems in the plane give the solution of the lattice-like covering problem: Determine those centrally symmetric convex bodies whose translated copies (with respect to a fixed lattice L) cover the space and have minimal surface area.

     

Affinely Regular Polygons as Extremals of Area Functionals

For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.     

Sharp convex Lorentz–Sobolev inequalities

New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L ^p Minkowski problem. New L ^p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.     

The spherical conjecture in Minkowski geometry

We show that the shapes of convex bodies containing m translates of a convex body K, so that their Minkowskian surface area is minimum, tends for growing m to a convex body L.Received: 7 January 2002     

Some large deviation results for sparse random graphs

Summary. We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected. We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently unknown characterisation, which is obtained by embedding the random graph in a random directed graph. The results demonstrate that, at this scaling, the properties `connected' and `contains no isolated vertices' are not asymptotically equivalent. (At the threshold probability they are asymptotically equivalent.) Received: 14 November 1996 / In revised form: 15 August 1997     

A new affine invariant for polytopes and Schneider's projection problem

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.

     

An isoperimetric inequality for convex polygons and convex sets with the same symmetrals

Suppose that two distinct plane convex bodies have the same Steiner symmetrals about a finite number n of given lines. Then we obtain an upper bound for the measure of their symmetric difference. The bound is attained if, and only if, the directions of the lines are equally spaced and the bodies are two regular concentric polygons, with n sides, each obtained from the other by rotation through an angle /n. This result follows from a new isoperimetric inequality for convex polygons.     

Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points

Summary. A general formula is proved, which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points. This formula is applied to improve some well-known results in geometric probability. For example, an estimate, which was established for a special case by L. C. G. Rogers, is obtained with the best possible bound, and an asymptotic formula due to A. Rényi and R.␣Sulanke is extended to an asymptotic expansion. Received: 21 May 1996     

Moment inequalities and central limit properties of isotropic convex bodies

The object of our investigations are isotropic convex bodies , centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies – specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body K as above as a probability space and taking , we define random variables on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions . We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to -convergence and -convergence of the corresponding densities. Received: 21 March 2001 / in final form: 17 October 2001 / Published online: 4 April 2002     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

New higher-order equiaffine invariants

We introduce new affine invariants for smooth convex bodies. Some sharp affine isoperimetric inequalities are established for the new invariants. Partially supported by an NSERC grant and an FRDP grant. Partially supported by an NSF grant, an FRG-NSF grant and a BSF grant.     

Positively curved graphs

This paper consists of two halves. In the first half of the paper, we consider real-valued functions f whose domain is the vertex set of a graph G and that are Lipschitz with respect to the graph distance. By placing a uniform distribution on the vertex set, we treat as a random variable. We investigate the link between the isoperimetric function of G and the functions f that have maximum variance or meet the bound established by the subgaussian inequality. We present several results describing the extremal functions, and use those results to (a) resolve a conjecture by Bobkov, Houdré, and Tetali characterizing the extremal functions of the subgaussian inequality of the odd cycle and (b) provide a construction that gives a partial negative answer to a question by Alon, Boppana, and Spencer on the relationship between maximum variance functions and the isoperimetric function of product graphs. While establishing a discrete analogue of the curved Brunn-Minkowski inequality for the discrete hypercube, Ollivier and Villani suggested several avenues for research. We resolve them in the second half of the paper as follows:

• They propose that a bound on t-midpoints can be obtained by repeated application of the bound on midpoints, if the original sets are convex. We construct a specific example where this reasoning fails, and then prove our construction is general by characterizing the convex sets in the discrete hypercube.
• A second proposed technique to bound t-midpoints involves new results in concentration of measure. We follow through on this proposal, with heavy use on results from the first half of the paper.
• We show that the curvature of the discrete hypercube is not positive or zero, but we also give a result indicating that it may satisfy a weaker version of curvature.