Archives of Inequalities and Applications xx (200x)xxx-xxx LOWER BOUNDS FOR NATURAL PSEUDODISTANCES VIA SIZE FUNCTIONS

Let us consider two C closed homeomorphic manifolds M, N and two C functions φ : M → R, ψ : N → R, called measuring functions. The natural pseudodistance d between the pairs (M, φ), (N , ψ) is defined as the infimum of Θ(f) def = maxP∈M |φ(P ) − ψ(f(P ))|, as f varies in the set of all homeomorphisms from M onto N . In this paper we show that size functions allow us to get a lower bound for d. Furthermore, we prove that this lower bound can be assumed equal either to |c′ − c′′| or to 12 |c′ − c′′|, where c′, c′′ are two suitable critical values of the measuring functions. AMS (MOS) Subject Classification.Primary 58C05; Secondary 53C15.