On the equivalence of modes of convergence for log-concave measures

An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note, we explore the extent to which different notions of distance between probability measures are comparable for log-concave distributions. Our results imply that weak convergence of isotropic log-concave distributions is equivalent to convergence in total variation, and is further equivalent to convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note

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

Nest use is influenced by the positions of nests and drinkers in aviaries

The influence of the nest location and the placement of nipple drinkers on nest use by laying hens in a commercial aviary was assessed. Twenty pens in a laying hen house were equipped with the same commercial aviary system, but the pens differed in the nest location and the placement of nipple drinkers. Nests were placed along the walls in 10 pens, and nipple drinkers were installed in front of the nests in 5 of these pens. The other 10 pens were equipped with nests placed on a tier within the aviary (integrated nests). Nipple drinkers were installed in front of the nests in 5 of these pens. A total of 225 Lohmann Selected Leghorns were housed per pen. The hens were offered 4 nests per pen: 2 facing the service corridor of the laying hen house and 2 facing the outdoor area. The numbers of nest eggs and mislaid eggs were counted daily per pen. At 25, 36, and 43 wk of age, the nest platforms were videotaped and the behavior of laying hens in front of the nests was analyzed. The nest location affected the stationary and locomotive behaviors in front of the nests. Hens in front of the integrated nests and the nests with drinkers displayed more stationary behaviors than hens in front of wall-placed nests or nests without drinkers. No difference in the number of nest eggs could be detected, but the integration of the nests inside the aviary led to a more even distribution of hens while nest searching. In the pens with wall-placed nests, significantly more hens laid eggs in the nests at the wall near the service corridor than at the wall near the outdoor area. Due to this imbalance, crowding in front of the preferred nests occurred and pushing and agonistic interactions on the nest platforms were significantly more frequent. Placement of nipple drinkers in front of nests had no effect on the number of eggs laid in those nest

Remarks on the Central Limit Theorem for Non-Convex Bodies

In this note, we study possible extensions of the Central Limit Theorem for non-convex bodies. First, we prove a Berry-Esseen type theorem for a certain class of unconditional bodies that are not necessarily convex. Then, we consider a widely-known class of non-convex bodies, the so-called p-convex bodies, and construct a counter-example for this class

Stability for Borell-Brascamp-Lieb inequalities

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.Comment: to appear in GAFA Seminar Note

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

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

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

Review: Assessment of completeness of reporting in intervention studies using livestock: an example from pain mitigation interventions in neonatal piglets

Accurate and complete reporting of study methods, results and interpretation are essential components for any scientific process, allowing end-users to evaluate the internal and external validity of a study. When animals are used in research, excellence in reporting is expected as a matter of continued ethical acceptability of animal use in the sciences. Our primary objective was to assess completeness of reporting for a series of studies relevant to mitigation of pain in neonatal piglets undergoing routine management procedures. Our second objective was to illustrate how authors can report the items in the Reporting guidElines For randomized controLled trials for livEstoCk and food safety (REFLECT) statement using examples from the animal welfare science literature. A total of 52 studies from 40 articles were evaluated using a modified REFLECT statement. No single study reported all REFLECT checklist items. Seven studies reported specific objectives with testable hypotheses. Six studies identified primary or secondary outcomes. Randomization and blinding were considered to be partially reported in 21 and 18 studies, respectively. No studies reported the rationale for sample sizes. Several studies failed to report key design features such as units for measurement, means, standard deviations, standard errors for continuous outcomes or comparative characteristics for categorical outcomes expressed as either rates or proportions. In the discipline of animal welfare science, authors, reviewers and editors are encouraged to use available reporting guidelines to ensure that scientific methods and results are adequately described and free of misrepresentations and inaccuracies. Complete and accurate reporting increases the ability to apply the results of studies to the decision-making process and prevent wastage of financial and animal resources