  We treat problems of fair division, their various interconnections, and their
relations to Sperner's lemma and the KKM theorem as well as their variants. We
prove extensions of Alon's necklace splitting result in certain regimes and
relate it to hyperplane mass partitions. We show the existence of fair cake
division and rental harmony in the sense of Su even in the absence of full
information. Furthermore, we extend Sperner's lemma and the KKM theorem to
(colorful) quantitative versions for polytopes and pseudomanifolds. For
simplicial polytopes our results turn out to be improvements over the earlier
work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma.
Moreover, our results extend the work of Musin on quantitative Sperner-type
results for PL manifolds.
