Minimum recession-compatible subsets of closed convex sets

A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimal with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn’s Lemma. An application of this result to the error bound theory in optimization is presented.     

Monotone translation invariant valuations on convex bodies

Archiv der Mathematik -     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

p-Pseudomeasures and closed subgroups

LetPM _p (G) be the space of allp-pseudomeasures on a locally compact groupG. We show the existence of a conditional expectation fromPM _p (G) ontoPM _p (H) whereH is a closed normal subgroup ofG. As an application we give a new proof of the fact thatH is a set ofp-synthesis inG; we also get an inequality involving the operator norm of bounded measures onG. Moreover, in analogy with a theorem of Reiter, we obtain a result concerning the closed ideals of the Figà-Talamanca Herz algebra ofG.     

Weak convergence of an empirical process indexed by the closed convex subsets ofI 2

Summary LetI ² be the unit cube of andX _i be independentI ²-valued random variables that are distributed according to Lebesgue-measure. IfS is the set of closed convex subsets ofI ² we consider the process _n (A) _AS,where .It is proved that this process suitably normalized converges in a suitable weak sense to a Gaussian process.     

A property of planar convex bodies

It is proved that from each interior point of a planar convex body emanate three distinct vectors terminating on the boundary, the sum of any two of which also terminates in the boundary. Some other related results are obtained.     

A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection P _n of points on the plane contains two such that any circle containing them contains n/c elements of P _n , c a constant. We prove: Let Φ be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S _i , S _j of Φ such that any set S' homothetic to S that contains them contains n/c elements of Φ, c a constant (S' is homothetic to S if , where λ is a real number greater than 0 and is a vector of ). Our proof method is based on a new type of Voronoi diagram, called the ``closest covered set diagram' based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Φ of n disjoint convex sets in such that for any nonempty subset Φ _H of Φ there is a sphere S _H containing all the elements of Φ _H , and no other element of Φ. Received December 7, 1994, and in revised form March 25, 1996.     

Integral geometry of translation invariant functionals,II: The case of general convex bodies

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.     

Fuzzy invariant subgroups and fuzzy ideals

The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals, and to prove some fundamental properties. In particular, it will give a characteristic of a field by a fuzzy ideal.     

Perfect subsets of invariant CA-sets

The familiar theorem that any ₂ ¹ (a)-set X of real numbers (where a is a fixed real parameter) not containing a perfect kernel necessarily satisfies the condition X L[a] is extended to a wider class of sets, with countable ordinals allowed as additional parameters in ₂ ¹ (a)-definitions.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 334–338.Original Russian Text Copyright © 2005 by V. G. Kanovei, V. A. Lyubetskii.This revised version was published online in April 2005 with a corrected issue number.     

A property of convex sets and its application

Translated from Matematicheskie Zametki, Vol. 44, No. 1, pp. 89–99, July, 1988.