Controlling Lipschitz functions
نویسندگان
چکیده
Given any positive integers m and d, we say the a sequence of points (xi)i∈I in Rm is Lipschitz-d-controlling if one can select suitable values yi (i ∈ I) such that for every Lipschitz function f : Rm → Rd there exists i with |f(xi)−yi| < 1. We conjecture that for every m ≤ d, a sequence (xi)i∈I ⊂ Rm is d-controlling if and only if sup n∈N |{i ∈ I : |xi| ≤ n}| nd =∞. We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be d-controlling. We also prove the conjecture for m = 1.
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