  This paper consists of two halves.
  In the first half of the paper, we consider real-valued functions $f$ whose
domain is the vertex set of a graph $G$ and that are Lipschitz with respect to
the graph distance. By placing a uniform distribution on the vertex set, we
treat $f$ as a random variable. We investigate the link between the
isoperimetric function of $G$ and the functions $f$ that have maximum variance
or meet the bound established by the subgaussian inequality. We present several
results describing the extremal functions, and use those results to resolve:
(A) a conjecture by Bobkov, Houdr\'e, and Tetali characterizing the extremal
functions of the subgaussian inequality of the odd cycle, and (B) a conjecture
by Alon, Boppana, and Spencer on the relationship between maximum variance
functions and the isoperimetric function of product graphs.
  While establishing a discrete analogue of the curved Brunn-Minkowski
inequality for the discrete hypercube, Ollivier and Villani suggested several
avenues for research. We resolve them in second half of the paper as follows.
(1) They propose that a bound on $t$-midpoints can be obtained by repeated
application of the bound on midpoints, if the original sets are convex. We
construct a specific example where this reasoning fails, and then prove our
construction is general by characterizing the convex sets in the discrete
hypercube. (2) A second proposed technique to bound $t$-midpoints involves new
results in concentration of measure. We follow through on this proposal, with
heavy use on results from the first half of the paper. (3) We show that the
curvature of the discrete hypercube is not positive or zero.
