Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures

Bobkov, Sergey G.

doi:10.1214/aop/1022874820

Cited by 154 publications

(241 citation statements)

References 30 publications

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“…An alternative approach to localization for proving isoperimetric inequalities was developed by Bobkov [18] in the Euclidean setting. Bobkov's approach was extended by Barthe [6] and subsequently by Barthe and Kolesnikov [8].…”

Section: Estimating D Chementioning

confidence: 99%

On the role of convexity in isoperimetry, spectral gap and concentration

2009

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We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, GromovMilman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are "on-average" Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the "worst" subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne-Weinberger, Li-Yau, KannanLovász-Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry-Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature-dimension condition of BakryEmery.

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Section: Estimating D Chementioning

confidence: 99%

On the role of convexity in isoperimetry, spectral gap and concentration

2009

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“…In [6], [7], [8], an important observation was made that several functional inequalities that hold for Gaussian, such as Poincare and logarithmic Sobolev inequalities as well as reverse entropy power inequalities, also hold for a random variable X whose density function p(x) is log concave, i.e.,…”

Section: Arxiv:151008341v4 [Csit] 2 Sep 2017mentioning

confidence: 99%

Bounds on Variance for Unimodal Distributions

Chung

Sadler

Hero

2017

IEEE Trans. Inform. Theory

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We show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy power. Combining this bound with the wellknown entropy power lower bound on variance, we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the scaled entropy power. As differential entropy decreases, the variance is sandwiched between two exponentially decreasing functions in the differential entropy. This establishes that for the subclasses of unimodal distributions, the differential entropy can be used as a surrogate for concentration of the distribution.

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“…This area was initiated by Soyster (1973) and it was further developed by Ben-Tal and Nemirovski (1998, 1999, Goldfarb and Iyengar (2003) and Bertsimas and Sim (2004). A robust policy can be defined in different ways.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Also note that the family of log-concave distributions, as defined by distributions where the log of the density function is concave, is a rather extensive family, which includes uniform, normal, logistic, extreme-value, chi-square, chi, exponential, laplace, among others. For further reference, a deeper understanding of log-concave distributions and their properties can be seen in Bagnoli and Bergstrom (1989) and Bobkov (1999).…”

Section: Sampling Based Optimizationmentioning

confidence: 99%

Dynamic Pricing Through Sampling Based Optimization

Lobel

Perakis

2010

SSRN Journal

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In this paper we develop an approach to dynamic pricing that combines ideas from data-driven and robust optimization to address the uncertain and dynamic aspects of the problem. In our setting, a firm offers multiple products to be sold over a fixed discrete time horizon. Each product sold consumes one or more resources, possibly sharing the same resources among different products. The firm is given a fixed initial inventory of these resources and cannot replenish this inventory during the selling season. We assume there is uncertainty about the demand seen by the firm for each product and seek to determine a robust and dynamic pricing strategy that maximizes revenue over the time horizon. While the traditional robust optimization models are tractable, they give rise to static policies and are often too conservative. The main contribution of this paper is the exploration of closed-loop pricing policies for different robust objectives, such as MaxMin, MinMax Regret and MaxMin Ratio. We introduce a sampling based optimization approach that can solve this problem in a tractable way, with a confidence level and a robustness level based on the number of samples used. We will show how this methodology can be used for data-driven pricing or adapted for a random sampling optimization approach when limited information is known about the demand uncertainty.Finally, we compare the revenue performance of the different models using numerical simulations, exploring the behavior of each model under different sample sizes and sampling distributions.

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Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures

Cited by 154 publications

References 30 publications

On the role of convexity in isoperimetry, spectral gap and concentration

On the role of convexity in isoperimetry, spectral gap and concentration

Bounds on Variance for Unimodal Distributions

Dynamic Pricing Through Sampling Based Optimization

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