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.
Calculation of Minimum Spanning Tree Edges Lengths using Gromov--Hausdorff Distance
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abstract
In the present paper we show how one can calculate the lengths of edges of a minimum spanning tree constructed for a finite metric space, in terms of the Gromov-Hausdorff distances from this space to simplices of sufficiently large diameter. Here by simplices we mean finite metric spaces all of whose nonzero distances are the same. As an application, we reduce the problems of finding a Steiner minimal tree length or a minimal filling length to maximization of the total distance to some finite number of simplices considered as points of the Gromov-Hausdorff space.
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math.MG 3years
2019 3verdicts
UNVERDICTED 3representative citing papers
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.
Extends prior Gromov-Hausdorff distance results to simplexes from compact metric spaces to all bounded ones via partition geometry.
citing papers explorer
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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.
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Gromov--Hausdorff Distance to Simplexes
Extends prior Gromov-Hausdorff distance results to simplexes from compact metric spaces to all bounded ones via partition geometry.