Lately there has been a lot of interest on applications of the ham sandwich theorem and related results. There is a bunch of lecture notes and surveys that touch upon the subject. I dont know of any that touches the most recent developments and includes some applications to both combinatorial geometry and functional analysis. Is there a good reference in which they are summarized? The following is a list of the applications that I know of. Do you know of other applications? Heuristically, what is a good principle to recognize problems in which these theorems might lead to a solution?
Applications of the ham sandwich
Alon-Akiyama disjoint rainbows theorem. Alon-West, Goldberg-West. Necklaces for the thieves theorem. Barany-Valtr, Pach. Same type lemma. Matousek, Agarwal-Erickson. Geometric Range Search. Gromov-Milman, Concentration of measure for uniformly convex bodies (localization technique).
Polynomial Ham Sandwhich
Guth's multilinear Kakeya estimate, Guth-Katz: Joints problem, Distinct distances problem. Tao-Solymosi, Matousek-Sharir-Kaplan Generalizations and new proofs of Szemeredi-Trotter, Sum product estimates, Existence of not too selfintersecting geometric tree of Chazelle and Welzl.
Center Point Theorem
Zivaljevic-Vrecica, Dolnikov. The Transversal Center Point Theorem. Thurston-Vavais-Miller-Teng. Separator theorem for planar graphs and for intersection graphs of metric euclidean balls with bounded overlap.
The Yao-Yao partition
Lehec. The Blashke-Santalo functional inequality. Alon-Pach-Radocic-Sharir-Pinchasi Semialgebraic same type lemma for graph-like relations. Fox-Gromov-Naor-Lafforgue-Pach, Bukh-Hubard. Semialgebraic same type lemma.
Gromov, Memarian. Tight bounds on the Waist of the sphere for smooth maps. Karasev and Volovikov generalized Gromov's convex equpartitions theorem to obtain a Waist theorem for targets other than $R^k$.
Sorry for the possible mistakes in the pseudocitations.