CONTROLLING LIPSCHITZ FUNCTIONS

Given any positive integers m and d, we say a sequence of points (x(i))(i is an element of I) in R-m is Lipschitz-d-controlling if one can select suitable values y(i)(i is an element of I) such that for every Lipschitz function f : R-m -> R-d there exists i with vertical bar f (x(i)) - Y-i vertical bar < 1. We conjecture that for every m <= d, a sequence (x(i))(i is an element of I )subset of R-m is d-controlling if and only if sup(n is an element of N)vertical bar{i is an element of I : vertical bar X-i vertical bar <= n}vertical bar/n(d) = infinity. 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.