Yu introduced property A in 2000 in his work on the Novikov conjecture as a means to guarantee a uniform embedding into Hilbert space. The class of groups and metric spaces with property A is vast and includes spaces with finite asymptotic dimension or finite decomposition complexity, among others. We reduce property A to a sequence of linear programming optimization problems on finite graphs. We explore the dual problem, which provides a means to show that a graph fails to have property A. As consequences, we examine the difference between graphs with expanders and graphs without property A, we recover theorems of Willett and Nowak concerning graphs without property A, and arrive at a natural notion of mean property A. This is joint work with Andrzej Nagórko, University of Warsaw.

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Passcode: A group is called an ________ group if it admits an invariant mean. (8 letters, lowercase)