We derive lower bounds for the size of simplicial covers of simplotopes,
which are products of simplices. These also serve as lower bounds for
triangulations of such polytopes, including triangulations with interior
vertices. We establish that a minimal triangulation of a product of two
simplices is given by a vertex triangulation, i.e., one without interior
vertices. For products of more than two simplices, we produce bounds for
products of segments and triangles.

In this paper we prove a new combinatorial theorem for labellings of trees,
and show that it is equivalent to a KKM-type theorem for finite covers of trees
and to discrete and continuous fixed point theorems on finite trees. This is in
analogy with the equivalence of the classical Sperner's lemma, KKM lemma, and
the Brouwer fixed point theorem on simplices. Furthermore, we use these ideas
to develop new KKM and fixed point theorems for infinite covers and infinite
trees.

In contrast to the classical cake-cutting problem (how to fairly divide a
desirable object), "chore division" is the problem of how to divide an
undesirable object. We develop the first explicit algorithm for envy-free chore
division among N people, a counterpart to the N-person envy-free cake-division
solution of Brams-Taylor (1995). This is accomplished by exploiting a notion of
"irrevocable advantage" for chores. We discuss the differences between
cake-cutting and chore division and additional problems encountered in chore
division.