Supports of semi-stable probability measures on locally convex spaces

LetE be a locally convex space. Let be an absolutely convexly tight Radom semi-stable probability measure onE with index 1<2 and Lévy measureM. The main result of this paper shows that the closed semigroup generated by the support ofM and the negative of the barycenter ofM restricted to a suitable compact subset ofE is a (closed) linear space ofE, and that the support of is a suitable translate of this linear space. This result complements a few known results concerning the supports of stable and semi-stable probability measures. In particular, it extends an analogous result proved recently for the support of -stable probability measures 1<2 (Ref. 4). Related results concerning the support of Radon semi-stable probability measures onE of index 0<<1 are also discussed.The research of this author was partially supported by AFSOR Grant No. 90-0168The research of this author was supported by KBN Grant, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

Separated totally convex spaces

In this paper we define for every totally convex space a suitable topology, the radial topology. We prove that a morphism in the category TC^sep of separated totally convex spaces is an epimorphism if and only if its image is dense in the radial topology, and that TC^sep is the full subcategory of TC generated by its Hausdorff objects. These results remain valid for finitely totally convex spaces when the radial topology is replaced by the distance-radial topology.Dedicated to Karl Stein     

A ball separation property of bounded convex sets in Hilbert spaces

In this paper we provide a ball separation property of bounded convex sets in a Hilbert space. As a consequence, we obtain a representation form of convex closures and two results about convex functionals.     

A dichotomy for the convex spaces of probability measures

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has ‘small’ local character in M or else M contains a measure of ‘large’ Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.     

On weak compactness of bounded sets in Banach and locally convex spaces

We investigate the compactness of one class of bounded subsets in Banach and locally convex spaces. We obtain a generalization of the Banach-Alaoglu theorem to a class of subsets that are not polars of convex balanced neighborhoods of zero. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp 731–739, June. 2000.     

Choquet simplices as spaces of invariant probability measures on post-critical sets

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak^∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f   is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential −ln|f^′|$-\ln |f^{\prime }|$.     

Epimorphisms of separated totally convex spaces

We give characterizations of the epimorphisms in the categories of open and separated totally convex and (pre)separated absolutely convex spaces.     

Topological classification of spaces of probability measures for co-analytic sets

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 10–19, January, 1994.     

Semidefinite representation of convex sets

Let be a semialgebraic set defined by multivariate polynomials g _i (x). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S _i ) be the boundary of S (resp. S _i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of , does there exist an LMI representable set Ŝ in some higher dimensional space whose projection down onto equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g _i (x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ ^T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g _i (x) is either sos-concave ( − ∇² g _i (x) = W(x) ^T W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S _i is either sos-convex or poscurv-convex (S _i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g _i (x) ≠ 0 on ∂ S _i ∩ S), then S is SDP representable. This also holds for S _i for which ∂ S _i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).