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

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

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

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

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

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

Fetal heart rate and fetal heart rate variability in Lipizzaner broodmares

Monitoring fetal heart rate (FHR) and fetal heart rate variability (FHRV) helps to understand and evaluate normal and pathological conditions in the foal. The aim of this study was to establish normal heart rate reference values for the ongoing equine pregnancy and to perform a heart rate variability (HRV) time-domain analysis in Lipizzaner mares. Seventeen middle- and late-term (days 121–333) pregnant Lipizzaner mares were examined using fetomaternal electrocardiography (ECG). The mean FHR (P = 0.004) and the standard deviation of FHR (P = 0.012) significantly decreased during the pregnancy. FHR ± SD values decreased from 115 ± 35 to 79 ± 9 bpm between months 5 and 11. Our data showed that HRV in the foal decreased as the pregnancy progressed, which is in contrast with the findings of earlier equine studies. The standard deviation of normal-normal intervals (SDNN) was higher (70 ± 25 to 166 ± 108 msec) than described previously. The root mean square of successive differences (RMSSD) decreased from 105 ± 69 to 77 ± 37 msec between the 5th and 11th month of gestation. Using telemetric ECG equipment, we could detect equine fetal heartbeat on day 121 for the first time. In addition, the large differences observed in the HR values of four mare-fetus pairs in four consecutive months support the assumption that there might be ‘high-HR’ and ‘low-HR’ fetuses in horses. It can be concluded that the analysis of FHR and FHRV is a promising tool for the assessment of fetal well-being but the applicability of these parameters in the clinical setting and in studs requires further investigation

Human Analogue Safe Haven Effect of the Owner : Behavioural and Heart Rate Response to Stressful Social Stimuli in Dogs

The secure base and safe haven effects of the attachment figure are central features of the human attachment theory. Recently, conclusive evidence for human analogue attachment behaviours in dogs has been provided, however, the owner’s security-providing role in danger has not been directly supported. We investigated the relationship between the behavioural and cardiac response in dogs (N = 30) while being approached by a threatening stranger in separation vs. in the presence of the owner, presented in a balanced order. Non-invasive telemetric measures of heart rate (HR) and heart rate variability (HRV) data during the threatening approaches was compared to periods before and after the encounters. Dogs that showed distress vocalisation during separation (N = 18) and that growled or barked at the stranger during the threatening approach (N = 17) were defined as behaviourally reactive in the given situation. While characteristic stress vocalisations were emitted during separations, the absence of the owner did not have an effect on dogs’ mean HR, but significantly increased the HRV. The threatening approach increased dogs’ mean HR, with a parallel decrease in the HRV, particularly in dogs that were behaviourally reactive to the encounter. Importantly, the HR increase was significantly less pronounced when dogs faced the stranger in the presence of the owner. Moreover, the test order, whether the dog encountered the stranger first with or without its owner, also proved important: HR increase associated with the encounter in separation seemed to be attenuated in dogs that faced the stranger first in the presence of their owner. We provided evidence for human analogue safe haven effect of the owner in a potentially dangerous situation. Similarly to parents of infants, owners can provide a buffer against stress in dogs, which can even reduce the effect of a subsequent encounter with the same threatening stimuli later when the owner is not present

Familiarity bias and physiological responses in contagious yawning by dogs support link to empathy

In humans, the susceptibility to yawn contagion has been theoretically and empirically related to our capacity for empathy. Because of its relevance to evolutionary biology, this phenomenon has been the focus of recent investigations in nonhuman species. In line with the empathic hypothesis, contagious yawning has been shown to correlate with the level of social attachment in several primate species. Domestic dogs (Canis familiaris) have also shown the ability to yawn contagiously. To date, however, the social modulation of dog contagious yawning has received contradictory support and alternative explanations (i.e., yawn as a mild distress response) could explain positive evidence. The present study aims to replicate contagious yawning in dogs and to discriminate between the two possible mediating mechanisms (i.e., empathic vs. distress related response). Twenty-five dogs observed familiar (dog’s owner) and unfamiliar human models (experimenter) acting out a yawn or control mouth movements. Concurrent physiological measures (heart rate) were additionally monitored for twenty-one of the subjects. The occurrence of yawn contagion was significantly higher during the yawning condition than during the control mouth movements. Furthermore, the dogs yawned more frequently when watching the familiar model than the unfamiliar one demonstrating that the contagiousness of yawning in dogs correlated with the level of emotional proximity. Moreover, subjects’ heart rate did not differ among conditions suggesting that the phenomenon of contagious yawning in dogs is unrelated to stressful events. Our findings are consistent with the view that contagious yawning is modulated by affective components of the behavior and may indicate that rudimentary forms of empathy could be present in domesticated dogs