Log-concave functionsSurface area measureThe classic Riesz representation theorem characterizes all linear and increasing functionals on the space C c ( X ) of continuous compactly supported functions. A geomet
Integration of log-concave functions log-concave在积分下不能保证log-concave的性质,但是假设f:R^n\times R^m\to R是log-concave的,那么 g(x)=\int f(x,y) dy 对于\forall x\in R^n是log-concave的 上述结论的推论,假设f(x),g(x)均为R^n上的log-concave function,定义卷积 (f*g)(x)=\int...
A classification of \\({ext {SL}}(n)\\) and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are...
Moments of Log-Concave FunctionsGaussian Random VectorsWe prove a sharp moment inequality for a log-concave or a log-convex function, on Gaussian random vectors. As an application we take a stability result for the classical logarithmic Sobolev inequality of L. Gross \\cite{Gross} in the case...
restrictive, we note that many “reasonable” functions belong to this family. We discuss this assumption in Section II, and point out some interesting properties of -log-concave functions. In Section III, we prove an inequality stating that the mea- ...
Using Bregman distances, we present new, proper convex Bayes cost functions for which the MAP estimator is the Bayes estimator. We complement this finding by results that correct further common misconceptions about MAP estimates. In total, we aim to rehabilitate MAP estimates in linear inverse ...
In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $n$ can estimate a log-concave density with respect to the squared Hellinger ...
ofdensityfunctionstothoseofreliabilityfunctions,failurerates,andthemono- tonicityofthe“mean-residual-lifetimefunction.”Wedefinethe“mean-advantage- over-inferiorsfunction”fortruncateddistributionsandrelatemonotonicityofthis functiontolog-concavityorlog-convexityoftheprobabilitydensityfunctionand ...
Here we extend these inequalities to log-concave functions.A.M FinkMathematical and Computer ModellingA.M. FINK, Hadamard's inequality for log-concave functions, Math. Comput. Modelling, 32(5-6) (2000), 625-629, Boundary value problems and related top- ics.Fink AM: Hadamard's inequality ...
log-concave functionsThe Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is affirmative if n≤4 and negative if n≥5.In this paper,we ...