Optimal Design of Sensors via Geometric Criteria

Abstract

We consider a convex set Omega and look for the optimal convex sensor omega subset Omega of a given measure that minimizes the maximal distance to the points of Omega . This problem can be written as follows inf{dH(ω,Ω)||ω|=candω⊂Ω},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\inf \\{d^H(\\omega ,\\Omega ) \\ |\\ |\\omega |=c\\ \ ext {and}\\ \\omega \\subset \\Omega \\}, \\end{aligned}$$\\end{document}where cin (0,|Omega |),d^H being the Hausdorff distance. We show that the parametrization via the support functions allows us to formulate the geometric optimal shape design problem as an analytic one. By proving a judicious equivalence result, the shape optimization problem is approximated by a simpler minimization problem of a quadratic function under linear constraints. We then present some numerical results and qualitative properties of the optimal sensors and exhibit an unexpected symmetry breaking phenomenon.