  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 $d$-cube $[0,1]^d$ in which points on
opposite faces must be given different colors, and for any $\varepsilon>0$,
there is an $\ell_\infty$ $\varepsilon$-ball which contains points of at least
$(1+\frac{\varepsilon}{1+\varepsilon})^d$ different colors, (so in particular,
at least $(1+\frac{2}{3}\varepsilon)^d$ different colors for all sensible
$\varepsilon\in(0,\frac12]$).
