Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR ⁿ . The Minkowski functional ∥ ∥ _U ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i (L,U)i=1,...,n, are defined in the usual way. Let ℒ _n be the family of all lattices inR ⁿ . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ _n andu∈R ⁿ /(L+U), somev∈L ^* with ∥v∥ _V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U ⁰), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl _p ⁿ , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U ⁰) are less thanCn logn for some numerical constantC.     

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.     

On theL p-bounds for maximal functions associated to convex bodies inR n

The dimension-freeL ²-maximal inequality for convex symmetric bodies obtained in [2] is extended forp>3/2.     

Polar reciprocal convex bodies

The minimum of the product of the volume of a symmetric convex bodyK and the volume of the polar reciprocal body ofK relative to the center of symmetry is attained for the cube and then-dimensional crossbody. As a consequence, there is a sharp upper bound in Mahler’s theorem on successive minima in the geometry of numbers. The difficulties involved in the determination of the minimum for unsymmetricK are discussed. Reserch partially supported by NSF Grant GP-27960. An erratum to this article is available at .     

Lattice inequalities for convex bodies and arbitrary lattices

Two isoperimetric inequalities with lattice constraints for arbitrary lattices are proved, where the last successive minimum of the lattice is used. The results generalize previous results by Hadwiger et al. for the special lattice ^d to general lattices.     

A limiting Lagrangian for infinitely constrained convex optimization inR n

For convex optimization inR ⁿ,we show how a minor modification of the usual Lagrangian function (unlike that of the augmented Lagrangians), plus a limiting operation, allows one to close duality gaps even in the absence of a Kuhn-Tucker vector [see the introductory discussion, and see the discussion in Section 4 regarding Eq. (2)]. The cardinality of the convex constraining functions can be arbitrary (finite, countable, or uncountable).In fact, our main result (Theorem 4.3) reveals much finer detail concerning our limiting Lagrangian. There are affine minorants (for any value 0<1 of the limiting parameter ) of the given convex functions, plus an affine form nonpositive onK, for which a general linear inequality holds onR ⁿAfter substantial weakening, this inequality leads to the conclusions of the previous paragraph.This work is motivated by, and is a direct outgrowth of, research carried out jointly with R. J. Duffin.This research was supported by NSF Grant No. GP-37510X1 and ONR Contract No. N00014-75-C0621, NR-047-048. This paper was presented at Constructive Approaches to Mathematical Models, a symposium in honor of R. J. Duffin, Pittsburgh, Pennsylvania, 1978. The author is grateful to Professor Duffin for discussions relating to the work reported here.The author wishes to thank R. J. Duffin for reading an earlier version of this paper and making numerous suggestions for improving it, which are incorporated here. Our exposition and proofs have profited from comments of C. E. Blair and J. Borwein.     

An analytical expression and an algorithm for the volume of a convex polyhedron inR n

An analytical expression for the volume of the convex polyhedron {x¦Ax?b} is given. Based on a simple recursive identity, it yields an efficient algorithm. Redundant constraints can be detected.     

The mixed area of a convex body and its polar reciprocal

Half the vector sum of a convex body and its polar reciprocal with respect to a unit sphereE containsE. A consequence of this is: The mixed area of a plane convex body and its polar reciprocal with respect toE is minimized by circles concentric withE. This work was supported in part by a grant from the National Science Foundation, NSF-G 19838. The author is indebted to the referee for fruitful comments.     

Inequalities for convex sequences and nondecreasing convex functions

In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing conve     

Centrally symmetric convex bodies and distributions II

In continuation of a previous work we study the generating distributions of centrally symmetric convex bodies and obtain some more geometric formulas and new characterizations of zonoids and generalized zonoids.     

Lower estimate of the isoperimetric deficit of convex domains inR n in terms of asymmetry

For convex bodiesD inR ⁿ it is shown that the isoperimetric deficit ofD is minorized by a constant times the square of thebarycentric asymmetry (D) ofD. Here (D) is defined as the volume ofDB _D divided by the volume ofD, whereB _D denotes the ball centred at the barycentre ofD and having the same volume asD.Dedicated to the memory of Børge Jessen     

A criterion for the affine equivalence of cell complexes inR d and convex polyhedra inR d+1

A criterion is given that decides, for a convex tilingC ofR ^d , whetherC is the projection of the faces in the boundary of some convex polyhedronP inR ^d+1. Its applicability is restricted neither by the properties nor by the dimension ofC. It turns out to be conceptually simpler than other criteria and allows the easy examination of various classes of cell complexes. In addition, the criterion is constructive, that is, it can be used to constructP provided it exists.Research was supported by the Austrian Fonds zur Foerderung der wissenschaftlichen Forschung.     

Random walks and an O*(n5) volume algorithm for convex bodies

Given a high dimensional convex body K⊆ℝⁿ by a separation oracle, we can approximate its volume with relative error ε, using O*(n⁵) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use “rounding” followed by a multiphase Monte-Carlo (product estimator) technique. Both parts rely on sampling (generating random points in K), which is done by random walk. Our algorithm introduces three new ideas: the use of the isotropic position (or at least an approximation of it) for rounding; the separation of global obstructions (diameter) and local obstructions (boundary problems) for fast mixing; and a stepwise interlacing of rounding and sampling. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 1–50, 1997