A note on measures of nonconvexity

We define the Hausdorff measure of nonconvexity β(C) of a nonempty bounded subset C of a Banach space X as the Hausdorff distance of C to the family of all the nonempty convex bounded subsets of X. We compare the measure β with the Eisenfeld-Lakshmikantham measure of nonconvexity α and prove that the two measures are equivalent (β≤α≤2β), but in general they are different.     

Weak-closure and polarization constant by Gaussian measure

Let X be a Banach space and (x_n) ì X{(x_n) \subset X} . We study under which condition on ||x _n || the set {x _n ; n ≥ 0} is weakly closed. Our results depend on the geometry of X.     

A note on reflexivity and nonconvexity

In this short note we prove that a Banach space X is reflexive if, and only if, the Eisenfeld–Lakshmikantham measure of nonconvexity in X satisfies the Cantor property. Using this characterization, some results in best approximation and fixed point theory for reflexive Banach spaces are generalized by removing convexity requirements.     

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.     

Essential and inessential points of local nonconvexity

LetS be a closed connected subset of a Hausdorff linear topological space,Q the points of local nonconvexity ofS, E the essential members ofQ, N the inessential. IfS～Q is connected, then the following are true: Theorem 1.If Q ?Øis countable, then S is planar. Theorem 2.If Q is finite and nonempty, then cardE≧cardN+1. Theorem 3.If SυR ² and N is infinite, then E is infinite.     

Stability for the Minkowski measure of convex domains of constant width

In 1988, H. Groemer gave a stability theorem for the area of convex domains of constant width. In this paper, we obtain a stability theorem for the well-known Minkowski measure of asymmetry for convex domains of constant width.     

The normalization constant of a certain invariant measure on GL n ( )

The ratio of the Tamagawa measure and a certain invariant measure on the group GL_n() is computed, where is the adèle of a division algebra D over a global field. An explicit formula of the ratio is described in terms of the special values of the zeta function of D. This formula yields (i) an explicit lower bound of the Hermite–Rankin constant _n,m(D) of D and (ii) an explicit asymptotic behavior of the distribution of rational points on Brauer–Severi variety.Mathematics Subject Classification (2000): Primary 11R52, Secondary 11H50     

On the degree and separability of nonconvexity and applications to optimization problems

We study qualitative indications for d.c. representations of closed sets in and functions on Hilbert spaces. The first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity. We show that a closed set is weakly closed if this indication is finite. Using this result we can prove the solvability of nonconvex minimization problems. By duality a minimization problem on a feasible set in which this indication is low, can be reduced to a quasi-concave minimization over a convex set in a low-dimensional space. The second indication is the separability which can be incorporated in solving dual problems. Both the index of nonconvexity and the separability can be characteristics to “good” d.c. representations. For practical computation we present a notion of clouds which enables us to obtain a good d.c. representation for a class of nonconvex sets. Using a generalized Caratheodory’s theorem we present various applications of clouds.     

Procedures for calculating the nonconvexity measures of a plane set

The geometry of nonconvex sets is analyzed. The measure of nonconvexity of a closed set that has the sense of an angle is considered. Characteristic manifolds of nonconvex sets are constructed. Procedures for calculating the measure of nonconvexity are proposed for a class of plane sets.     

A note on the extremal bodies of constant width for the Minkowski measure

In a previous paper, we showed that for all convex bodies K of constant width in ${\mathbb{R}^n, 1 \leq {\rm as}_\infty(K) \leq \frac{n+\sqrt{2n(n+1)}}{n+2}}$ , where as_∞(·) denotes the Minkowski measure of asymmetry, with the equality holding on the right-hand side if K is a completion of a regular simplex, and asked whether or not the completions of regular simplices are the only bodies for the equality. A positive answer is given in this short note.     

Points of local nonconvexity and sets which are almost starshaped

Let Sø be a bounded connected set in R ², and assume that every 3 or fewer lnc points of S are clearly visible from a common point of S. Then for some point p in S, the set A{s : s in S and [p, s] S} is nowhere dense in S. Furthermore, when S is open, then S in starshaped.     

On nonconvexity of the solution set of a system of linear interval equations

We give necessary and sufficient conditions for the solution set of a system of linear interval equations to be nonconvex and derive some consequences.     

On sets having finitely many points of local nonconvexity and propertyP m

If a pointq ofS has the property that each neighborhood ofq contains pointsx andy such that the segmentxy is not contained byS, q is called a point of local nonconvexity ofS. LetQ denote the set of points of local nonconvexity ofS. Tietze’s well known theorem that a closed connected setS in a linear topological space is convex ifQ=φ is generalized in the result:If S is a closed set in a linear topological space such that S ∼ Q is connected and |Q|=n<∞,then S is the union of n+1or fewer closed convex sets. Letk be the minimal number of convex sets needed in a convex covering ofS. Bounds fork in terms ofm andn are obtained for sets having propertyP _m and |Q|=n.     

Points of local nonconvexity,clear visibility,and starshaped sets in Rd

Let S be a compact, connected, locally starshaped set in R^d, S not convex. For every point of local nonconvexity q of S, define A_q to be the subset of S from which q is clearly visible via S. Then ker S = {conv A_q: q lnc S}. Furthermore, if every d+1 points of local nonconvexity of S are clearly visible from a common d-dimensional subset of S, then dim ker S = d.     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

The subgaussian constant and concentration inequalities

We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to various graph constants. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube. Research supported in part by NSF Grant No. DMS-0405587 and by EPSRC Visiting Fellowship. Research supported in part by NSF Grant No. DMS-9803239, DMS-0100289. Research supported in part by NSF Grant No. DMS-0401239.     

The Gabriel-Roiter measure

The first Brauer-Thrall conjecture asserts that algebras of bounded representation type have finite representation type. This conjecture was solved by Roiter in 1968. The induction scheme which he used in his proof prompted Gabriel to introduce an invariant which we propose to call Gabriel-Roiter measure. This invariant is defined for any finite length module and it will be studied in detail in this paper. Whereas Roiter and Gabriel were dealing with algebras of bounded representation type only, it is the purpose of the present paper to demonstrate the relevance of the Gabriel-Roiter measure for algebras in general, in particular for those of infinite representation type.     

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008