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We prove that any fat, subanalytic compact subset of $\mathbb R^N$ possesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic) sets like N-dimensional polynomial or analytic graph domains or polynomial and analytic polyhedrons. (Here a good source of references is the recent paper (Piazzon and Vianello, East J Approx 16(4):389---398, 2010).) We also show that an infinitely differentiable map f from a compact set Q in $\mathbb R^N$ onto a Markov compact set K in $\mathbb C^l$ (l驴驴驴N) transforms a (weakly) admissible mesh in Q onto a (weakly) admissible mesh in K, which extends a result of Piazzon and Vianello (East J Approx 16(4):389---398, 2010) for analytic maps in case Q is a subset of $\mathbb R^N$ . Versions for $\mathcal C^k$ maps with sufficiently large k are also given.

Nearly optimal meshes in subanalytic sets