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In 1977, Ahlswede and Katona proposed the following isoperimetric problem: find a set S of n points in {0,1}k whose average Hamming distance is minimal—or equivalently find an n-vertex subgraph of the hypercube Qk whose average distance is minimal. We report on some recent results and conjecture that S can be chosen to be the set of all points in {0,1}k that are distance at most r from some point...
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