Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
Ramón J. Aliaga, Chris Gartland, Colin Petitjean, A. Procházka
Abstract
We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a compact metric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a dual space if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , pure 1-unrectifiability is actually equivalent to some well-known Banach space properties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -finite Hausdorff 1-measure.