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In this paper we show that the equivalences between certain properties ofclosed subanalytic sets proved by E. Bierstone and P. Milman in \cite{[BM-1]}hold for closed sets definable in quasianalytic o-minimal structures. Inparticular we prove that uniform Chevalley estimate implies a stratification bythe diagram of initial exponents and further, Zariski semicontinuity of thediagram of initial exponents. We also show that the stratification by thediagram implies Zariski semicontinuity of Hilbert-Samuel function.

On the geometric and differential properties of closed sets definable in quasianalytic structures