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Capacité analytique et le problème de Painlevé

Hervé Pajot (2003/2004)

Séminaire Bourbaki

Le problème de Painlevé consiste à trouver une caractérisation géométrique des sous-ensembles du plan complexe qui sont effaçables pour les fonctions holomorphes bornées. Ce problème d’analyse complexe a connu ces dernières années des avancées étonnantes, essentiellement grâce au dévelopement de techniques fines d’analyse réelle et de théorie de la mesure géométrique. Dans cet exposé, nous allons présenter et discuter une solution proposée par X. Tolsa en termes de courbure de Menger au problème...

Capacités invariantes extrémales

Michel Talagrand (1978)

Annales de l'institut Fourier

On étudie certains cônes de mesures ≥ 0 sur un espace localement compact, qui sont invariantes par l’action continue d’un groupe localement compact G , cette étude étant centrée sur les génératrices extrémales de ces cônes. On dégage d’abord un type très simple d’action continue où l’on décrit complètement la situation. On dégage ensuite une classe d’actions (contenant par exemple l’action de shift de Bernoulli sur { 0 , 1 } N ) qui ne sont pas du type précédent, et que l’on étudie en grand détail. Le résultat...

Cardinality of some convex sets and of their sets of extreme points

Zbigniew Lipecki (2011)

Colloquium Mathematicae

We show that the cardinality of a compact convex set W in a topological linear space X satisfies the condition that ℵ ₀ = . We also establish some relations between the cardinality of W and that of extrW provided X is locally convex. Moreover, we deal with the cardinality of the convex set E(μ) of all quasi-measure extensions of a quasi-measure μ, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of extrE(μ).

Category bases.

Detlefsen, M., Szymański, Andrzej (1993)

International Journal of Mathematics and Mathematical Sciences

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q − 1 restricted to ∂ E , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q − 1 ( ∂ E ) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo...

Characterization of local dimension functions of subsets of ℝ d

L. Olsen (2005)

Colloquium Mathematicae

For a subset E ⊆ ℝ d and x ∈ ℝ d , the local Hausdorff dimension function of E at x is defined by d i m H , l o c ( x , E ) = l i m r ↘ 0 d i m H ( E ∩ B ( x , r ) ) where d i m H denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

Characterization of optimal shapes and masses through Monge-Kantorovich equation

Guy Bouchitté, Giuseppe Buttazzo (2001)

Journal of the European Mathematical Society

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

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