2 research outputs found
Neighborhood Variants of the KKM Lemma, Lebesgue Covering Theorem, and Sperner's Lemma on the Cube
We establish a "neighborhood" variant of the cubical KKM lemma and the
Lebesgue covering theorem and deduce a discretized version which is a
"neighborhood" variant of Sperner's lemma on the cube. The main result is the
following: for any coloring of the unit -cube in which points on
opposite faces must be given different colors, and for any ,
there is an -ball which contains points of at least
different colors, (so in particular,
at least different colors for all sensible
).Comment: 18 pages plus appendices (30 pages total), 3 figure