Mean width inequalities for symmetric Wulff shapes

We establish the mean width inequalities for symmetric Wulff shapes by a direct approach.We also yield the dual inequality along with the equality conditions.These new inequalities have Barthe’s mean width inequalities for even isotropic measures and its dual form as special cases.     

Mean square inequalities for chords of convex sets

This paper is concerned with establishing lower bounds for the integrals of the square of the lengths of area and perimeter bisecting chords of planar convex sets. The results obtained provide verification of two recent conjectures of Lutwak. When combined with the known upper bounds for these integrals they yield the classical isoperimetric inequality. The main proof technique involves estimation of the winding numbers of the locus of the midpoints of the chords concerned.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

Mean inequalities in certain Banach function spaces

Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.     

Geometric inequalities for a class of exponential measures

Using -ellipsoids we prove versions of the inverse Santaló inequality and the inverse Brunn-Minkowski inequality for a general class of measures replacing the usual volume on . This class contains in particular the Gaussian measure on .

     

Mean time exit and isoperimetric inequalities for minimal submanifolds of NxR

Based on Markvorsen and Palmer's work on mean time exit andisoperimetric inequalities, we establish a slightly better isoperimetricinequality and mean time exit estimates for minimal submanifoldsof N x when N is non-positively curved. We prove isoperimetricinequalities for submanifolds with tamed second fundamentalform in Hadamard spaces with bounded sectional curvature. Weuse mean time exit functions to show that spherically symmetricmanifolds with geodesic spheres with exponential volume growthhave a positive first eigenvalue.     

Clark formula and logarithmic Sobolev inequalities for Bernoulli measures

Using a Clark formula for the predictable representation of random variables in discrete time and adapting the method presented in [Electron. Commun. Probab. 2 (1997) 71–81] in the Brownian case, we obtain a proof of modified and L¹ logarithmic Sobolev inequalities for Bernoulli measures. We also prove a bound that improves these inequalities as well as the optimal constant inequality of [J. Funct. Anal. 156 (2) (1998) 347–365]. To cite this article: F. Gao, N. Privault, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Moment inequalities and central limit properties of isotropic convex bodies

The object of our investigations are isotropic convex bodies , centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies – specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body K as above as a probability space and taking , we define random variables on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions . We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to -convergence and -convergence of the corresponding densities. Received: 21 March 2001 / in final form: 17 October 2001 / Published online: 4 April 2002     

Mean dual and harmonic cross-sectional measures

Summary The mean dual cross-sectional measures are introduced. They are shown to satisfy a cyclic inequality similar to that satisfied by the cross-sectional measures (Quermassintegrale). A new representation of the dual cross-sectional measures is used to obtain inequalities relating the mean dual cross-sectional measures and the harmonic cross-sectional measures (Harmonische Quermassintegrale) of Hadwiger. An inequality between the volume and the harmonic cross-sectional measures of a convex body is presented. An inequality stronger than the Urysohn inequality (the harmonic Urysohn inequality) is proven. Strengthened versions of other inequalities previously obtained by the author are also presented. Entrata in Redazione il 13 luglio 1977.     

Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups

We study coercive inequalities in Orlicz spaces associated to the probability measures on finite- and infinite-dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities.     

Sharp inequalities for geometric maximal operators associated with general measures

We determine the best constants in the weak-type (p, p) and L ^p estimates for geometric maximal operator on (?, µ). It is also shown that in higher dimensions such inequalities fail to hold.     

Exponential inequalities for dynamical measures of expanding maps of the interval

 We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval. As an immediate corollary we obtain a concentration inequality. We apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure. Received: 14 August 2001 / Revised version: 13 February 2002 / Published online: 1 July 2002     

Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity

We show that the averaged response of random isotropic Cauchy elastic material can be described analytically. It leads to a higher gradient model with explicit expressions for the dependence on the second derivatives of the mean field. A subsequent penalty formulation coincides with a linear elastic micro-stretch model with specific choice of constitutive parameters, depending only on the average cut-off length (the internal characteristic length scale L_c > 0). Thus the microstretch displacement field can be interpreted as an approximated mean field response for these parameter ranges. The mean field free energy in this micro-stretch formulation is not uniformly pointwise positive, nevertheless, the model is well posed.      

Discontinuities of approximating measures on constant width sets

SiaK un insieme piano ad ampiezza costante δ, ?K la sua frontiera ev _t (K) la misura approssimante la misura unidimensionale di Hausdorff μ¹(K), ottenuta con ricoprimento chiusi di diametro non superiore att. E′ noto che l'unica discontinuità perv _t (K) e perv _t (?K) si può avere pert=δ. In questo scritto si approfondisce lo studio di queste discontinuità e si dimostra tra l'altro che i salti in δ div _t (K) ev _t (?K) sono uguali.     

Mean width of random polytopes in a reasonably smooth convex body

Let K be a convex body in and let X_n=(x₁,…,x_n) be a random sample of n independent points in K chosen according to the uniform distribution. The convex hull K_n of X_n is a random polytope in K, and we consider its mean width W(K_n). In this article, we assume that K has a rolling ball of radius >0. First, we extend the asymptotic formula for the expectation of W(K)−W(K_n) which was earlier known only in the case when ∂K has positive Gaussian curvature. In addition, we determine the order of magnitude of the variance of W(K_n), and prove the strong law of large numbers for W(K_n). We note that the strong law of large numbers for any quermassintegral of K was only known earlier for the case when ∂K has positive Gaussian curvature.