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Four-point CAT(0) condition
The definition of a metric space guarantees that if we pick three points from it, we can draw a triangle in the Euclidean plane, with vertices labeled so that the side lengths are equal to the...
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Re: “How many sides does a circle have?”
The post is inspired by this story told by JDH at Math.SE. My third-grade son came home a few weeks ago with similar homework questions: How many faces, edges and vertices do the following have? cube...
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Pentagrams and hypermetrics
The Wikipedia article Metric (mathematics) offers a plenty of flavors of metrics, from common to obscure: ultrametric, pseudometric, quasimetric, semimetric, premetric, hemimetric and pseudoquasimetric...
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Polygonal inequalities: beyond the triangle
(Related to previous post but can be read independently). The triangle inequality, one of the axioms of a metric space, can be visualized by coloring the vertices of a triangle by red and blue. The...
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Infinite beatitude of non-existence: a journey into Nothingland
In the novella Flatland by Edwin A. Abbott, the Sphere leads the Square “downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions”: I caught these words,...
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Graphical convergence
The space of continuous functions (say, on ) is usually given the uniform metric: . In other words, this is the smallest number such that from every point of the graph of one function we can jump to...
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Binary intersection property, and not fixing what isn’t broken
A metric space has the binary intersection property if every collection of closed balls has nonempty intersection unless there is a trivial obstruction: the distance between centers of two balls...
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Rough isometries
An isometry is a map between two metric spaces which preserves all distances: for all . (After typing a bunch of such formulas, one tends to prefer shorter notation: , with the metric inferred from...
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Weak convergence in metric spaces
Weak convergence in a Hilbert space is defined as pointwise convergence of functionals associated to the elements of the space. Specifically, weakly if the associated functionals converge to pointwise....
View ArticleMeasure-distanced subsets of an interval
Suppose is a bounded metric space in which we want to find points at safe distance from one other: for all . Let be the greatest value of for which this is possible (technically, the supremum which...
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