Convexity without convex combinations

Separation theorems play a central role in the theory of Functional Inequalities. The importance of Convex Geometry has led to the study of convexity structures induced by Beckenbach families. The aim of the present note is to replace recent investigations into the context of an axiomatic setting, for which Beckenbach structures serve as models. Besides the alternative approach, some new results (whose classical correspondences are well-known in Convex Geometry) are also presented.     

A characterization of isometries on an open convex set

Let X be a real Hilbert space with dim X ≥ 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f, from an open convex subset of X into Y, has a contractive distance ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry.     

Representation and index theory for C1-algebras generated by commuting isometries

We discuss here representation and Fredholm theory for C¹-algebras generated by commuting isometries. More particularly, for n commuting isometries {V_j: 1 ? j ? n} on separable Hilbert space we give a representation resembling the well-known representation for a single isometry. Our representation permits an analysis of the C¹-algebras $\text{Ol}$=$\text{Ol}$(V_j:1?j?n) generated by the {V_j}. The commutator ideal in $\text{Ol}$ is identified precisely and, under certain additional hypotheses, the Fredholm operators in $\text{Ol}$ are also precisely determined. Finally, we obtain formulas in terms of topological data for the index of Fredholm operators in some interesting algebras of the type $\text{Ol}$(V_j:1?j?n).     

Dominated Pesin theory: convex sum of hyperbolic measures

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures?To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index, such as the dominated splitting, is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class.We provide examples which indicate the importance of the domination assumption.     

Nekhoroshev estimates for commuting nearly integrable symplectomorphisms

In this paper, we prove the Nekhoroshev estimates for commuting nearly integrable symplectomorphisms. We show quantitatively how ? ^m symmetry improves the stability time. This result can be considered as a counterpart of Moser’s theorem [11] on simultaneous conjugation of commuting circle maps in the context of Nekhoroshev stability. We also discuss the possibility of Tits’ alternative for nearly integrable symplectomorphisms.     

A characterization of isometries on an open convex set,II

Let E ⁿ be an n-dimensional Euclidean space with n ≥ 2. In this paper, we generalize a classical theorem of Beckman and Quarles by proving that if a mapping, from an open convex subset Co of E ⁿ into E ⁿ , preserves a distance ρ, then the restriction of f to an open convex subset C _∞ of C ₀ is an isometry. This work was supported by 2007 Hongik University Research Fund.     

Regularity estimates for convex multifunctions

The main result of the paper contains an exact formula for the rate of regularity of a set-valued mapping with a convex graph. As a consequence we find an exact expression for the rate of regularity of set-valued mappings associated with so-called constraint systems. It turns out that the rate is equal to the upper bound of Robinson-type estimates over all norms in the graph space of the homogenized mapping majorizing the norm of the underlying space. We further introduce a concept of a perfectly regular mapping and find some criteria for perfect regularity.     

The group of isometries of a homogeneous convex irreducible self adjoint cone

Sunto In questo lavoro si considerano i coni aperti, convessi, omogenei, autoaggiunti ed irriducibili, V, di uno spazio vettoriale reale di dimensione finita. Si dimostra che il gruppo delle isometrie di V, su se stesso, rispetto alla metrica riemanniana canonica data dalla funzione caratteristica di V, è il prodotto diretto del gruppo degli automorfismi lineari di V col gruppo {Identità, }, ove è l'involuzione di V sul suo cono duale. Si prova, quindi, che ogni isometria di V, su se stesso, è la restrizione a V di un automorfismo olomorfo del dominio tubolare associato a V.     

Unbiased estimates and convex loss functions

Under fairly general assumptions it is proved that unbiased estimates of minimal variance are optimal in the class of unbiased estimates and with respect to any convex loss function. The analysis is extended to the case of matrix loss functions. A class of loss functions is introduced and studied that are universal for a given family of distributions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 34, pp. 40–52, 1974.     

Pointwise estimates for marginals of convex bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E⊂Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c>0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .     

Idempotency of linear combinations of three idempotent matrices, two of which are commuting

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67-78] who characterized all situations in which a linear combination P=c₁P₁+c₂P₂+c₃P₃, with c_i, i=1,2,3, being nonzero complex scalars and P_i, i=1,2,3, being nonzero complex idempotent matrices such that two of them, P₁ and P₂ say, are disjoint, i.e., satisfy condition P₁P₂=0=P₂P₁, is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P₁ and P₂ to the commutativity condition P₁P₂=P₂P₁.     

Some quantum estimates of Hermite-Hadamard inequalities for convex functions

In this study, based on a new quantum integral identity, we establish some quantum estimates of Hermite-Hadamard type inequalities for convex functions. These results generalize and improve some known results given in literatures.     

欧氏空间凸多面体极点凸组合表示定理的推广

使用切面技术、归纳法等证明了欧氏空间中凸集极点的存在性,进一步证明了空间中一般有界闭凸集(不只局限于凸多面体)中任意一点同样可表示为极点的凸组合.方法独到.     

Hessian estimates for convex solutions to quadratic Hessian equation

We derive Hessian estimates for convex solutions to quadratic Hessian equation by compactness argument.