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.
Borsuk, ¨Uber die Zerlegung einer n-dimensionalen Vollkugel in n- Mengen
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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.