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Many theorems in discrete geometry may be interpreted as relatives or combinatorial analogues of results on concentration of maps and measures. Configuration spaces of mixed combinatorial/geometric nature, such as arrangements of points, lines, convex polytopes, decorated trees, graphs, and partitions, often arise via the Configuration Space/Test Maps scheme, as spaces parameterizing feasible candidates for the solution of a problem in discrete geometry. One of the basic ideas is to link the key questions of social sciences dealing with fair allocations, such as the existence of certain Nash equilibria, equipartitions, or balanced configurations, with the existence of partitions of point sets satisfying some geometric constraints. In both contexts, combinatorial and geometric aspects of Fourier analysis on finite groups arise naturally.