On concentrators and related approximation constants

Pippenger (1977) [3] showed the existence of (6m,4m,3m,6)(6m,4m,3m,6)-concentrator for each positive integer mm using a probabilistic method. We generalize his approach and prove existence of (6m,4m,3m,5.05)(6m,4m,3m,5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44.5 (established by Kalton and Roberts in (1983) [2]) to 39. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in RdRd by linear functions and improve the estimate of this Whitney constant from 802 (proved by Brudnyi and Kalton in (2000) [1]) to 73.