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IMA Journal of Mathematical Control and Information 1999 16(3):207-220; doi:10.1093/imamci/16.3.207
© 1999 by Institute of Mathematics and its Applications
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Shapes and measures

A. FAKHARZADEH and J. E. RUBIO

Department of Mathematics, Shahid Chamran University of Ahwaz Ahwaz, Iran
Department of Applied Mathematical Studies, University of Leeds Leeds LS2 9JT, UK

It is a well-known fact that a measurable set a shape can be considered as a measure; the aim of this work is to solve an optimal-shape problem in such a way that it also answers the question of whether measures can be considered as shapes. This paper introduces a new method for solving problems of optimal shape design; by a process of embedding, the problem is replaced by another in which we seek to minimize a linear form over a subset of the product of two measure spaces defined by linear equalities. This minimization is global, and the theory allows us to develop a computational method which enables us to find the solution by finite-dimensional linear programming. The nearly optimal pair (C, dC) is obtained via the optimal pair of measures by an approximation procedure. It is sometimes necessary to apply a standard minimization algorithm, because of some limitations in the accuracy. Some examples are presented.


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