Logarithmically-concave moment measures I

We discuss a certain Riemannian metric, related to the toric Kahler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in R^n. We use this metric in order to bound the second derivatives of the solution to the toric Kahler-Einstein equation, and in order to obtain spectral-gap estimates similar to those of Payne and Weinberger.Comment: 27 page

Do Minkowski averages get progressively more convex?

Abstract Let us define, for a compact set A ⊂ R n , the Minkowski averages of A: We study the monotonicity of the convergence of A(k) towards the convex hull of A, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures of convergence. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, we prove that a strong form of monotonicity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing. Résumé Les moyennes de Minkowski deviennent-elles progressivement plus convexes ? Pour tout ensemble compact A ⊂ R n , définissons ses moyennes de Minkowski par Nousétudions la monotonie de la convergence de A(k) vers l'enveloppe convexe de A, mesurée par la distance de Hausdorff, le déficit volumique et par l'indice de non-convexité de Schneider. Pour le déficit volumique, nous démontrons que la propriété de monotonie n'est pas satisfaite en général, réfutant ainsi une conjecture de Bobkov, Madiman et Wang. Pour l'index de non-convexité de Schneider, nous montrons une propriété renforcée de monotonie tandis que pour la distance de Hausdorff, nousétablissons que la suite est strictement décroissanteà partir d'un certain rang. Version française abrégée L'objectif de cette note est d'annoncer et de démontrer une partie des résultats obtenus dans [3] qui portent sur l'étude de la monotonie de la suite (A(k)) k≥1 définie en (1), mesuréeà travers différentes mesures de non-convexité. Intuitivement, les ensembles A(k) deviennent de plus en plus convexes au fur età mesure que k croît. Cette intuition est précisée dans désigne le déficit volumique d'un ensemble compact de R n . Ici, Vol n représente la mesure de Lebesgue dans R n et conv(A) désigne l'enveloppe convexe de A. Nous réfutons cette conjecture en exhibant un contre-exemple explicite en dimension supérieure ouégaleà 12. Le contre-exemple est la réunion de deux ensembles convexes inclus dans des sous-espaces de dimension (presque) moitié de l'espace ambiant (voir De manière analogueà la conjecture de Bobkov-Madiman-Wang, nousétudions la monotonie de la suite (c(A(k))) k≥1 , où c est l'index de non-convexité de Schneider [6] défini par c(A) := inf{λ ≥ 0 : A + λ conv(A) est convexe}. Contrairement au déficit volumique, la suite (c(A(k))) est strictement décroissante,à moins que A(k) soit déjà convexe. Plus précisément nous montrons que pour tout ensemble compact A de R n et tout k ∈ N

Optimal Concentration of Information Content For Log-Concave Densities

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543, 2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with more general sufficient conditions for equality in Theorem 2.3. Also some minor corrections and added reference

Two remarks on generalized entropy power inequalities

This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed log-concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on

Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

We obtain new principles for transferring log-Sobolev and Spectral-Gap inequalities from a source metric-measure space to a target one, when the curvature of the target space is bounded from below. As our main application, we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of various conservative spin system models, consisting of non-interacting and weakly-interacting particles, constrained to conserve the mean-spin. When the self-interaction is a perturbation of a strongly convex potential, this partially recovers and partially extends previous results of Caputo, Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg and Yau. When the self-interaction is only assumed to be (non-strongly) convex, as in the case of the two-sided exponential measure, we obtain sharp estimates on the system's spectral-gap as a function of the mean-spin, independently of the size of the system.Comment: 57 page

Identification of Quantitative Trait Loci responsible for embryonic lethality in mice assessed by ultrasonography.

Recurrent Spontaneous Abortion (RSA) is a frequent pathology affecting 1 to 5% of couples. In approximately 50 % of cases, the aetiology is unknown suggesting a subtle interaction between genetic and environmental factors. Previous attempts to describe genetic factors using the candidate gene approach have been relatively unsuccessful due to the physiological, cellular and genetic complexity of mammalian reproduction. Indeed, fertility can be considered as a quantitative feature resulting from the interaction of genetic, epigenetic and environmental factors. Herein, we identified Quantitative Trait Loci (QTL) associated with diverse embryonic lethality phenotypes and the subsequent embryonic resorption in 39 inter-specific recombinant congenic mice strains, using in vivo ultrasound bio-microscopy. The short chromosomal intervals related to the phenotypes will facilitate the study of a restricted number of candidate genes which are potentially dysregulated in patients affected by RSA

A Rac/Cdc42 exchange factor complex promotes formation of lateral filopodia and blood vessel lumen morphogenesis

During angiogenesis, Rho GTPases influence endothelial cell migration and cell-cell adhesion; however it is not known whether they control formation of vessel lumens, which are essential for blood flow. Here, using an organotypic system that recapitulates distinct stages of VEGF-dependent angiogenesis, we show that lumen formation requires early cytoskeletal remodelling and lateral cell-cell contacts, mediated through the RAC1 guanine nucleotide exchange factor (GEF) DOCK4. DOCK4 signalling is necessary for lateral filopodial protrusions and tubule remodelling prior to lumen formation, whereas proximal, tip filopodia persist in the absence of DOCK4. VEGF-dependent Rac activation via DOCK4 is necessary for CDC42 activation to signal filopodia formation and depends on the activation of RHOG through the RHOG GEF, SGEF. VEGF promotes interaction of DOCK4 with the CDC42 GEF DOCK9. These studies identify a novel Rho-family GTPase activation cascade for the formation of endothelial cell filopodial protrusions necessary for tubule remodelling, thereby influencing subsequent stages of lumen morphogenesis

Microtubules as Platforms for Assaying Actin Polymerization In Vivo

The actin cytoskeleton is continuously remodeled through cycles of actin filament assembly and disassembly. Filaments are born through nucleation and shaped into supramolecular structures with various essential functions. These range from contractile and protrusive assemblies in muscle and non-muscle cells to actin filament comets propelling vesicles or pathogens through the cytosol. Although nucleation has been extensively studied using purified proteins in vitro, dissection of the process in cells is complicated by the abundance and molecular complexity of actin filament arrays. We here describe the ectopic nucleation of actin filaments on the surface of microtubules, free of endogenous actin and interfering membrane or lipid. All major mechanisms of actin filament nucleation were recapitulated, including filament assembly induced by Arp2/3 complex, formin and Spir. This novel approach allows systematic dissection of actin nucleation in the cytosol of live cells, its genetic re-engineering as well as screening for new modifiers of the process