Journal ArticleDOI
Convex set functions ind-space
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In this article, it was shown that a convex subset of Euclidean d-space R d is convex (concave) if the inequality F(OA + (1 O)B)Abstract:
Given subsets A and B of Euclidean d-space R a and 0 ~ 0, we set A + B -{x + Y l x E A, y E B} and OA = {Ox Ix 6 A }. Further given a convex subset g2 of R d we shall say that a set function F : 2 ~ \ {~} ~ [0, + ~ ] is convex (concave} if the inequality F(OA + (1 O)B) ~ Or(A} + (1 0 ) / ' ( B ) (>=.) holds for all It ~ A, B c_ D, and all 0 < 0 < 1. Here we shall s tudy such set functions of the special form given in the followingread more
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Book
Convex bodies : the Brunn-Minkowski theory
TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
Book
Lectures on Stochastic Programming: Modeling and Theory
TL;DR: The authors dedicate this book to Julia, Benjamin, Daniel, Natan and Yael; to Tsonka, Konstatin and Marek; and to the Memory of Feliks, Maria, and Dentcho.
Book ChapterDOI
On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
TL;DR: In this article, the authors extend the Prekopa-leindler theorem to other types of convex combinations of two positive functions and strengthen it by introducing the notion of essential addition.
Journal ArticleDOI
The Brunn-Minkowski inequality
TL;DR: In this article, the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications, is discussed; see Section 5.1.1 for a survey.
Journal ArticleDOI
The empirical content of the roy model
James J. Heckman,Bo E. Honoré +1 more
TL;DR: In this article, the authors consider the identifiability of the Roy model from data on earnings distributions and show that the normal theory version is identifiable without regressors or exclusion restrictions and with sufficient price variation, the model can be identified from multimarket data.
References
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Real and complex analysis
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Journal ArticleDOI
Distributions in Statistics: Continuous Multivariate Distributions.
Journal ArticleDOI