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On concentrators and related approximation constants
Pippenger ([Pippenger, 1977]) showed the existence of
-concentrator for each positive integer using a probabilistic
method. We generalize his approach and prove existence of
-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 (established in
[Kalton, Roberts, 1983]) to . We show a more direct connection of the
latter problem to the Whitney type estimate for approximation of continuous
functions on a cube in by linear functions, and improve the
estimate of this Whitney constant from (proved in [Brudnyi, Kalton,
2000]) to
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