Shapes and measures

It is a well-known fact that a measurable set a shape can beconsidered as a measure; the aim of this work is to solve anoptimal-shape problem in such a way that it also answers thequestion of whether measures can be considered as shapes. Thispaper introduces a new method for solving problems of optimalshape design; by a process of embedding, the problem is replacedby another in which we seek to minimize a linear form over asubset of the product of two measure spaces defined by linearequalities. This minimization is global, and the theory allowsus to develop a computational method which enables us to findthe solution by finite-dimensional linear programming. The nearlyoptimal pair (C, dC) is obtained via the optimal pair of measuresby an approximation procedure. It is sometimes necessary toapply a standard minimization algorithm, because of some limitationsin the accuracy. Some examples are presented.