Exact Gromov-Hausdorff distances are derived between arbitrary simplexes and 2-distance spaces, yielding a complete solution to the generalized Borsuk problem and expressions for graph clique cover and chromatic numbers.
Gromov--Hausdorff Distance to Simplexes
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abstract
Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The corresponding calculations essentially use geometry of partitions of these spaces. In the finite case, it gives the lengths of minimal spanning trees. A similar theory for compact metric spaces was worked out previously. In the present paper we generalize those results to any bounded metric space, and also, we simplify some proofs.
fields
math.MG 3years
2019 3verdicts
UNVERDICTED 3representative citing papers
The generalized Borsuk problem is solved by the criterion that a bounded metric space X admits an m-partition into smaller-diameter sets if and only if its Gromov-Hausdorff distance to an m-point simplex of smaller diameter is positive.
New closed-form expression for Gromov-Hausdorff distance between a simplex and a bounded metric space (under cardinality condition), extended to exact distance with ultrametric spaces.
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The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces
Exact Gromov-Hausdorff distances are derived between arbitrary simplexes and 2-distance spaces, yielding a complete solution to the generalized Borsuk problem and expressions for graph clique cover and chromatic numbers.
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Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes
The generalized Borsuk problem is solved by the criterion that a bounded metric space X admits an m-partition into smaller-diameter sets if and only if its Gromov-Hausdorff distance to an m-point simplex of smaller diameter is positive.
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The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces
New closed-form expression for Gromov-Hausdorff distance between a simplex and a bounded metric space (under cardinality condition), extended to exact distance with ultrametric spaces.