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Excerpt from Extreme Varieties, Concave Functions, and the Fixed Charge Problem This paper deals with various structural properties of convex sets and their application to the study of concave mappings. The investigation was prompted by a programming problem (the fixed charge problem) which involves the minimization of a concave function on a convex polyhedron. In the fixed charge problem it is not difficult to show (by an argument involving the form of the function and the polyhedral character of the domain) that the desired infimum is actually attained Oh the other hand it is not generally true that a concave function on a convex set attains its infimum, and it is natural to seek conditions under which this is so. The search for such conditions leads to the study of representations of convex sets similar to the Krein Milman theorem. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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