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We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal., Author(s): Ulrich Bauer [sup.1] , Claudia Landi [sup.2] , Facundo Mémoli [sup.3] Author Affiliations: (1) grid.6936.a, 0000000123222966, Department of Mathematics, Technical University of Munich (TUM), , Munich, Germany (2) grid.7548.e, [...]

MacMahon's Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighted voting games. We introduce a new generalized generating function that gives, as special cases, extensions of the generating functions of the Banzhaf, Shapley-Shubik, Banzhaf-Owen, symmetric coalitional Banzhaf, and Owen power indices. Our extensions involve any coalition formation related to a linear diophantine system and multiple voting games. In addition, we show that a combination of ideas from MPA and Clifford algebras is useful in constructing generating functions for coalition configuration power indices. Finally, a brief account on how to design voting systems via MPA is advanced. More precisely, we obtain new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not. Finding: This work was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq-Brazil) under [Grant 307211/2015-0]. Keywords: Banzhaf power index * Shapley-Shubik power index * coalitional power indices * coalition configuration * generating functions * linear diophantine system * Omega Calculus * Clifford algebra * weighted voting games * design of voting systems, 1. Introduction Power indices for weighted voting games (WVGs) have been a subject of recurrent interest (see, e.g., Felsenthal and Machover [25], Holler and Nurmi [33]). Two of the main [...]

We consider a nonatomic selfish routing model with independent stochastic travel times for each edge, represented by mean and variance latency functions that depend on edge flows. This model can apply to traffic in the Internet or in a road network. Variability negatively impacts packets or drivers by introducing jitter in transmission delays, which lowers quality of streaming audio or video, or by making it more difficult to predict the arrival time at destination. At equilibrium, agents may select paths that do not minimize the expected latency so as to obtain lower variability. A social planner, who is likely to be more risk neutral than agents because it operates at a longer time scale, quantifies social cost with the total expected delay along routes. From that perspective, agents may make suboptimal decisions that degrade long-term quality. We define the price of risk aversion (PRA) as the worst-case ratio of the social cost at a risk-averse Wardrop equilibrium to that where agents are risk neutral. This inefficiency metric captures the degradation of system performance caused by variability and risk aversion. For networks with general delay functions and a single source--sink pair, we first show upper bounds for the PRA that depend linearly on the agents' risk tolerance and on the degree of variability present in the network. We call these bounds structural, as they depend on the structure of the network. To get this result, we rely on a combinatorial proof that employs alternating paths that are reminiscent of those used in max-flow algorithms. For series-parallel graphs, the PRA becomes independent of the network topology and its size. Next, we provide tight and asymptotically tight lower bounds on the PRA by showing a family of structural lower bounds, which grow linearly with the number of nodes in the graph and players' risk aversion. These are tight for graph sizes that are powers of 2. After that, by focusing on restricting the set of allowable mean latency and variance functions, we derive functional bounds on the PRA that are asymptotically tight and depend on the allowed latency functions but not on the topology. The functional bounds match the price-of-anarchy bounds for congestion games multiplied by an extra factor that accounts for risk aversion. Finally, we turn to the mean-standard deviation user objective--a much more complex model of risk aversion because the cost of a path is nonadditive over edge costs--and provide tight bounds for instances that admit alternating paths with one or two forward subpaths. Funding: Part of this work was done while the first two authors were visiting the Simons Institute for the Theory of Computing at Berkeley and the third author had a faculty position at Universidad Torcuato Di Telia in Buenos Aires, Argentina. This research was supported by National Science Foundation [Grants CCF-1216103, CCF-1350823, and CCF-1331863], National Scientific and Technical Research Council (CON1CET), Argentina [Grant PIP 112-201201-00450CO], and Agenda Nacional de Promocion Cientifica y Tecnologica (ANPCyT), Argentina [Grant PICT-2012-1324]. Keywords: stochastic Wardrop game * risk aversion * price of anarchy, 1. Introduction A central question in decision making is how to make good decisions under uncertainty, particularly when decision makers are risk averse. Applications of crucial national importance, including alleviating [...]

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games. Funding: Financial support from the National Science Foundation [Grants SES-1155135, SES-1458992, and SES-1728853], U.S. Army Research Office [Grant MSN201957], and U.S. Air Force Office of Scientific Research [Grant FA9550-09-0538] is gratefully acknowledged. Keywords: sample path large deviations * evolutionary game theory * stochastic stability * Markov chains * potential games * congestion games, 1. Introduction Evolutionary game theory concerns the dynamics of aggregate behavior of populations of strategically interacting agents. To define these dynamics, one specifies the population size N, the game being [...]

By generalizing a recent result of Hong, Liu and Lai on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple k-node-connected digraphs. More generally, we solve the directed node-connectivity augmentation problem when the augmented digraph is degree-specified and simple. As for edge-connectivity augmentation, we solve the special case when the edge-connectivity is to be increased by one and the augmenting digraph must be simple. Funding: The research was supported by the Hungarian Scientific Research Fund--OTKA [Grant K109240]. The work of the first author was financed by a postdoctoral fellowship provided by the Hungarian Academy of Sciences. Keywords: degree-sequences * k-connected digraphs * connectivity augmentation, 1. Introduction There is an extensive literature of problems concerning degree sequences of graphs or digraphs with some prescribed properties such as simplicity or -connectivity. For example, Edmonds [8] characterized [...]

The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising k disjoint branchings [B.sub.i] each having a specified number [[mu].sub.i] of arcs. (B) As an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach. Funding: The research was supported by the Hungarian Scientific Research Fund--OTKA [Grant K109240]. The work of the first author was financed by a postdoctoral fellowship provided by the Hungarian Academy of Sciences. Keywords: supermodular arc-covering * packing branchings * term rank, 1. Introduction Network flow theory provides a basic tool to conveniently treat various graph characterization and optimization problems such as the degree-constrained subgraph problem in a bipartite graph (or bigraph, [...]

The multiple exchange property for matroid bases is generalized for valuated matroids and M-natural concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities condition of Gul and Stacchetti is, in fact, equivalent to the gross substitutes condition of Kelso and Crawford. Funding: This work was supported by The Mitsubishi Foundation, JST CREST [Grant JPMJCR14D2], and JSPS KAKENHI [Grant 26280004]. Keywords: discrete convex analysis * matroid * exchange property * combinatorial optimization, 1. Introduction Discrete convex analysis (Fujishige [5], Murota [17]) offers a general framework of discrete optimization, combining the ideas from submodular/matroid theory and convex analysis. It has found applications in [...]

Ryser's max term rank formula with graph theoretic terminology is equivalent to a characterization of degree sequences of simple bipartite graphs with a specific matching number. In a previous paper by the authors, a generalization was developed for the case when the degrees are constrained by upper and lower bounds. Here, two other extensions of Ryser's theorem are discussed. The first one is a matroidal model, while the second one settles the augmentation version. In fact, the two directions shall be integrated into one single framework. Funding: The research was supported by the Hungarian Scientific Research Fund-OTKA [Grant K109240]. The work of the first author was financed by a postdoctoral fellowship provided by the Hungarian Academy of Sciences. Keywords: matroids * term rank * bipartite matching * augmentation, 1. Introduction Ryser [13] derived a formula for the maximum term rank of a (0, l)-matrix with specified row sums and column sums. In graph theoretic terms, his theorem is [...]

We introduce an iterative method for solving linearly constrained optimization problems, whose nonsmooth nonconvex objective function is defined as the pointwise maximum of finitely many concave functions. Such problems often arise from reformulations of certain constraint structures (e.g., binary constraints, finite max-min constraints) in diverse areas of optimization. We state a local optimization strategy, which exploits piecewise-concavity of the objective function, giving rise to a linearized model corrected by a proximity term. In addition we introduce an approximate line-search strategy, based on a curvilinear model, which, similarly to bundle methods, can return either a satisfactory descent or a null-step declaration. Termination at a point satisfying an approximate stationarity condition is proved. We embed the local minimization algorithm into a Variable Neighborhood Search scheme and a Coordinate Direction Search heuristic, whose aim is to improve the current estimate of the global minimizer by exploring different parts of the feasible region. We finally provide computational results obtained on two sets of small- to large-size test problems. Keywords: nonsmooth optimization * minmax problems * nonconvex optimization * piecewise-concave, 1. Introduction We consider the following nonconvex nonsmooth optimization problem [mathematical expression not reproducible] (1) where [mathematical expression not reproducible] and we assume that * the component functions [f.sub.i](*), i[member [...]

We introduce concepts of metric regularity and metric subregularity of a positive-order for an implicit multifunction and provide new sufficient conditions for the implicit multifunctions to achieve the addressed properties. [...]

This work is concerned with finite-state irreducible Markov decision chains satisfying continuity-compactness requirements. It is supposed that the system is driven by a decision maker with utility function U, which, [...]

This work is devoted to extend several asymptotic results concerning repeated games with incomplete information on one side. The model we consider is a generalization of the classical model of [...]

In this work we construct subdivision schemes refining general subsets of [R.sup.n] and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane-Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally, we discuss the extension of the results obtained in metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties. Keywords Subdivision * Set-valued functions * Approximation * Symmetric difference metric Mathematics Subject Classification 26E25 * 41A65 * 68U99, 1 Introduction Approximation of set-valued functions (SVFs) has various potential applications in optimization, control theory, mathematical economics, medical imaging and more. The problem is closely related to the approximation of [...]

We discuss consistency of vanishingly smooth fictitious play, a strategy in the context of game theory, which can be regarded as a smooth fictitious play procedure, where the smoothing parameter [...]

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the dataset, for example by adding points or by applying functions to it. We show that, within this framework, one can prove a theorem analogous to one of Kleinberg (Becker et al. (eds.), NIPS, pp. 446-453, MIT Press, Cambridge, 2002), in which, for example, one obtains an existence and uniqueness theorem instead of a nonexistence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes, it is possible to construct richer families of clustering schemes that exhibit sensitivity to density. Keywords Clustering * Functoriality * Chaining effect * Single linkage * Data analysis Mathematics Subject Classification Primary 62H30 * Secondary 68T10, 1 Introduction Clustering techniques play a very central role in various parts of data analysis. They can give important clues to the structure of datasets, and therefore suggest results and [...]

Given a function f : E [right arrow] R with E [subset] [R.sup.n], we explain how to decide whether f extends to a [C.sup.m] function F on [R.sup.n]. If E is finite, then one can efficiently compute an F as above, whose [C.sup.m] norm has the least possible order of magnitude (joint work with B. Klartag).

A study investigating the continuity properties of effective functions is presented.

Byline: E. G. Goluzina (1) Abstract: The paper studies the region of values of the system {f(z.sub.1), f(z.sub.2),... , f(z.sub.n)} in the class T of functions f(z) = z + [a.sub.2]z.sup.2 + a- regular in the unit disk and satisfying the condition Im f(z) Im z &gt 0 for Im z a 0. Bibliography: 8 titles. Author Affiliation: (1) St.Petersburg Department, Steklov Mathematical Institute, St.Petersburg, Russia Article History: Registration Date: 01/02/2006 Received Date: 25/06/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 41--51.

Byline: A. V. Olesov (1) Abstract: For analytic functions satisfyingMeyman's majorization conditions, with the use of classical properties of conformal mappings exact inequalities complementing and strengthening the results of Akhiezer and Meyman are derived. As a corollary, for the modulus of the derivative of a rationally-trigonometric function a Bernstein type bound, which implies the result by Borwein, Erdelyi, and Zhang, is obtained. Bibliography: 10 titles. Author Affiliation: (1) Institute of Applied Mathematics, Far-Eastern Branch, Russian Academy of Sciences, Vladivostock, Russia Article History: Registration Date: 01/02/2006 Received Date: 19/04/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 155--173.

Byline: E. P. Golubeva (1) Abstract: Let S.sub.2(q) be the set of primitive forms in the space S.sub.2(I.sub.0(q)) of holomorpic I.sub.0(q)-cusp forms of weight 2. Let f a S.sub.2(q) and let L.sub.f(S) be the L-function of f(z). It is proved that the set {log L.sub.f(1), f a S.sub.2(q)} has a limit distribution function. The rate of convergence to this limit function is estimated. Bibliography: 10 titles. Author Affiliation: (1) Bonch-Bruevich State University for Telecommunications, St.Petersburg, Russia Article History: Registration Date: 01/02/2006 Received Date: 10/09/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 15--32.

We examine the effect of presmoothing functional data on estimating the dissimilarities among objects in a dataset, with applications to cluster analysis and other distance methods, such as multidimensional scaling and statistical matching. We prove that a shrinkage method of smoothing results in a better estimator of the dissimilarities among a set of noisy curves. For a model with independent noise structure, the smoothed-data dissimilarity estimator dominates the observed-data estimator. For a dependent-error model--often applicable when the functional data are measured nearly continuously over some domain--an asymptotic domination result is given for the smoothed-data estimator. A simulation study indicates the magnitude of improvement provided by the shrinkage estimator and examines its behavior for heavy-tailed noise structure. The shrinkage estimator presented here combines Stein estimation and basis function-based linear smoothers in a novel manner. Statisticians increasingly analyze sizable sets of functional data, and the results in this article are a useful contribution to the theory of the effect of presmoothing on functional data analysis. KEY WORDS: Cluster analysis; Dissimilarity measures; Distance methods; Multidimensional scaling; Shrinkage estimation; Smoothing; Statistical matching; Stein estimation.

Byline: A. Soldatov (1) Abstract: A survey on the theory of hyperanalytic functions in the sense of Douglis is presented. Some applications of hyperanalytic functions are also given, in particular, to the description of solutions of elliptic systems in the plane. Special attention is devoted to some systems arising in the plane elasticity theory and linearized Stokes system of hydromechanics. Author Affiliation: (1) Novgorod State University, Russia Article History: Registration Date: 11/01/2006 Article note: Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

Byline: G. Giorgadze (1) Abstract: We discuss a number of topics concerned with certain boundary-value problems in the context of generalized analytic functions. Solution of the classical Riemann-Hilbert problem and the linear conjugation problem for analytic functions is described in appropriate function classes and the same scheme is applied to generalized analytic functions and vectors. In particular, we describe solution of the Riemann-Hilbert problem for generalized analytic functions and obtain an explicit analytic presentation of monodromy matrices in the case of generalized analytic vectors. Author Affiliation: (1) Institute of Cybernetics, Georgian Academy of Sciences, Georgia Article History: Registration Date: 11/01/2006 Article note: Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

Byline: T. Hirsch (1), B. Martin (1) Abstract: We study modular subspaces corresponding to two deformation functors associated with an isolated singularity X .sub.0: the functor Def.sub.X.sub.0 of deformations of X .sub.0 and the functor Def.sub.X.sub.0 .sup.s of deformations with section of X .sub.0. After recalling some standard facts on the cotangent cohomology of analytic algebras and the general theory of deformations with section, we give several criteria for modularity in terms of the relative cotangent cohomology modules of a deformation. In particular, it is shown that the modular strata for the functors Def.sub.X.sub.0 and Def.sub.X.sub.0 .sup.s of quasi-homogeneous complete intersection singularities coincide. Then flatness conditions for the first cotangent cohomology modules of the deformation functors under consideration are compared. Author Affiliation: (1) Institut fur Mathematik, BTU Cottbus, Postfach 10 13 44, D-03046, Cottbus, Germany Article History: Registration Date: 11/01/2006 Article note: Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

Abstract: We consider a class of explicitly computed majorants that provide accuracy estimates for a solution to the Reissner-Mindlin plate problem. The proof is based on a generalized statement of the problem. It does not require any extra analysis of properties of controlled approximate solutions is independent of features of the finite element method for obtaining approximate solutions. Bibliography: 13 titles. Illustrations: 1 figure. Article History: Registration Date: 30/12/2005 Article note: Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 159--167.

Abstract: For an arbitrary nondegenerate convex compact set K a [].sup.m we construct analogs of Steklov functions. We establish estimates for the seminorms in the space of continuous functions on K in terms of various moduli of continuity. Some uniform approximation aggregates for functions of two variables are also studied. Bibliography: 11 titles. Article History: Registration Date: 30/12/2005 Article note: Dedicated to the memory of Professor of St.Petersburg University Garald Isidorovich Natanson Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 3--20.

Byline: N. V. Velichko (1) Keywords: function space; [lambda]-topology Abstract: This paper is devoted to the spaces C .sub.[lambda](X) of all continuous real-valued functions on X endowed with arbitrary [lambda]-topologies. This is a fairly complete survey of the results obtained by the author in the following domains of the theory of [lambda]-topologies: cardinal functions locally convex properties weak and strong topologies dual spaces lattices of [lambda]-topologies completeness. Author Affiliation: (1) Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, 16 ul. S. Kovalevskoj, Ekaterinburg, 620066, Russia Article History: Registration Date: 26/10/2005 Article note: Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 3--56, 2003.

Byline: A. P. Kombarov (1) Keywords: normality; [delta]-normality; F.sup.[sigma]-[delta]-normality; weak normality; weak normality over a class of spaces; -normality; pseudonormality; countable normality; D-normality; tightness; product; [summation over (term)]-product; [sigma]-product; space of closed subsets; power of a space; normal functor; countable compactness; compactness; paracompactness; p-space Abstract: Topological properties similar to normality are considered in subspaces of products and powers of topological spaces, of spaces of closed subsets, and of spaces having the form F (X), where F is a normal functor. Author Affiliation: (1) Department of General Topology and Geometry, Mechanics and Mathematics Faculty, Moscow State University, Leninskie gory, Moscow, 119992, Russia Article History: Registration Date: 26/10/2005 Article note: Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 57--98, 2003.

Byline: M. Borowiecka (1) Author Affiliation: (1) Technical University of Zielona Gora, Podgorna, 50, 65-246, Zielona Gora, Poland Article History: Registration Date: 09/10/2004

Byline: L. Lakatos (1) Author Affiliation: (1) Eotvos Lorand University, POB 32, H-1518, Budapest, Hungary Article History: Registration Date: 09/10/2004

Byline: V. V. Kapustin (1) Abstract: We study the following problem: to describe weights w on the unit circle such that the analytic and antianalytic subspaces of the corresponding weighted space .sup.Lp(w) have nonzero intersection. In the special case p=2, the problem is equivalent to the well-known problem on exposed points in H .sup.1. We show that the property in question is local, i.e., it depends on local behavior of the weight w at each point of the unit circle. Some necessary and sufficient conditions in terms of the Herglotz integrals are obtained. Bibliography: 6 titles. Author Affiliation: (1) St.Petersburg Department of the Steklov Mathematical Institute, Russia Article History: Registration Date: 18/10/2004

Abstract: Two families of functions constructed by a system of n scalar Muckenhoupt weights are studied. Criteria are given under which these families are unconditional bases. From the point of view of the spectral operator theory, the problem is reduced to the study of the structure of n-dimensional perturbations of the integration operator. Weighted estimates for the Hilbert transform in the spaces of vector-functions are applied to construct an operator mapping functions of the studied families to vector-valued rational functions. The concept of the Carleson series is used in the study of the following problem: when do vector-valued rational functions form an unconditional basis? Bibliography: 8 titles. Article History: Registration Date: 21/10/2004

Byline: T. S. Davydova (1), N. A. Shirokov (2) Abstract: Let I.sub.[alpha] (0 &lt [alpha] &lt 1) be the Holder class on the semi-axis [0,[infinity]). We characterize the class I.sub.[alpha] by rates of approximation of its functions by entire functions of order 1/2 belonging to a special class (similarly to the classical JacksonaBernstein theorem). Bibliography: 4 titles. Author Affiliation: (1) St.Petersburg State Pedagogical University, Russia (2) Russia Article History: Registration Date: 18/10/2004

Byline: V. I. Vasyunin (1) Abstract: It is well known that the Riemann hypothesis is equivalent to the following statement: the identity function belongs to the linear span in L .sup.2(0,1) of the family [ [alpha] /x ] - [alpha] [ 1/x ], 0 < [alpha] < 1. A step is presented in describing the set of all idempotents representable as a finite sum of functions of the form (*). Bibliography: 10 titles. Author Affiliation: (1) St.Petersburg Department of the Steklov Mathematical Institute, Russia Article History: Registration Date: 18/10/2004

Byline: A. V. Evdokimov (1), V. A. Zalgaller (2) Abstract: Let K.sub.0,K.sub.1,...,K.sub.m be nonempty compact sets in R.sup.n . Then the convex hulls .sub.i=0.sup.m (K.sub.i + r.sub.i) with r.sub.0=0 form a convex family of sets parametrized by [rho] = (r.sub.1,r.sub.m) [member of] R.sup.nm . For m=1, the volume Volconv([K.sub.0]aa(K.sub.1+r)) is a convex function of r [member of] R.sup.n . Bibliography: 5 titles. Author Affiliation: (1) St.Petersburg State University, Russia (2) St.Petersburg Department of the Steklov Mathematical Institute, Russia Article History: Registration Date: 22/10/2004