2024/01/17 by Mauricio Che, Fernando Galaz-García, Fernando Galaz‐García +3 · 1 citation
Computer Science · Mathematics · Engineering · #Computational Geometry and Mesh Generation #Mathematics and Applications #Advanced Numerical Analysis Techniques
paper · doi:10.1090/proc/16776
Given a metric pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e. a metric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a distinguished closed set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A⊂ X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , one may construct in a functorial way a pointed pseudometric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D Subscript normal infinity Baseline left-parenthesis upper X comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal D_∞ (X,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D Subscript normal infinity Baseline left-parenthesis upper X comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal D_∞ (X,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and obtain characterizations of their metrizability, completeness, separability, and geodesicity.