Let us consider two $C^1$ closed homeomorphic manifolds $mathcal{M}$, $mathcal{N}$ and two $C^1$ functions $varphi:{mathcal{M}}rightarrow mathbb{R}$, $psi:mathcal{N}rightarrow mathbb{R}$, called measuring functions. The natural pseudodistance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. 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 $frac{1}{2}|c'-c''|$, where $c'$, $c''$ are two suitable critical values of the measuring functions.

P. Donatini, P. Frosini (2004). Lower bounds for natural pseudodistances via size functions. ARCHIVES OF INEQUALITIES AND APPLICATIONS, 2(1), 1-12.

Lower bounds for natural pseudodistances via size functions

DONATINI, PIETRO;FROSINI, PATRIZIO
2004

Abstract

Let us consider two $C^1$ closed homeomorphic manifolds $mathcal{M}$, $mathcal{N}$ and two $C^1$ functions $varphi:{mathcal{M}}rightarrow mathbb{R}$, $psi:mathcal{N}rightarrow mathbb{R}$, called measuring functions. The natural pseudodistance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. 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 $frac{1}{2}|c'-c''|$, where $c'$, $c''$ are two suitable critical values of the measuring functions.
2004
P. Donatini, P. Frosini (2004). Lower bounds for natural pseudodistances via size functions. ARCHIVES OF INEQUALITIES AND APPLICATIONS, 2(1), 1-12.
P. Donatini; P. Frosini
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/12267
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