1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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The Brunn-Minkowski inequality

This paper clarifies and extends the classical Roy model of self selection and earnings inequality. The original Roy model, based on the assumption that log skills are normally distributed, is shown to imply that pursuit of comparative advantage in a free market reduces earnings inequality compared to the earnings distribution that would result if workers were randomly assigned to sectors. Aggregate log earnings are right skewed even if one sectoral distribution is left skewed. Most major implications of the log normal Roy model survive if differences in skills are log concave. However few implications of the model survive if skills are generated from more general distributions. We consider the identifiability of the Roy model from data on earnings distributions. The normal theory version is identifiable without regressors or exclusion restrictions. Sectoral distributions can be identified knowing only the aggregate earnings distribution. For general skill distributions, the model is not identified and has no empirical content. With sufficient price variation, the model can be identified from multimarket data. Cross-sectional variation in regressors can substitute for price variation in restoring empirical content to the Roy model.

The empirical content of the roy model

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 following

C. Borell

Papers

Convex set functions ind-space