| Summary: | <p> This dissertation contains a variety of results in quantitative combinatorial geometry, as well as applications to optimization and number theory. </p><p> We use Ehrhart theory, the study of the number of lattice points in polytopes, to prove a Rainbow Ramsey analogue of Richard Rado's 1933 result in Ramsey Theory. There were some conjectures as to what this analogue would be; they are disproven by our result. We also prove a few corollaries. </p><p> A portion of this dissertation contains new quantitative variants of classical convexity theorems: whereas the classical theorems have hypotheses and conclusions requiring certain sets to be nonempty, their quantitative variants require that the sets have a certain "size" (volume, diameter, etc). The three classical theorems we quantize are Carathéodory's Theorem, Helly's Theorem, and Tverberg's Theorem. The Helly portion contains new non-quantitative results as well. The Tverberg portion proves that any Helly-type theorem implies a corresponding Tverberg-type theorem. </p><p> A <i>maximal 1-lattice polytope</i> is a lattice polytope containing exactly one interior lattice point whose facets each contain at least one interior lattice point. In this dissertation, we bound the size of all maximal 1-lattice pyramids and prisms. These bounds may be used with a brute-force algorithm to quickly enumerate all maximal 1-lattice pyramids and prisms up to equivalence. </p><p> <i>S</i>-optimization, a generalization of continuous, integer, and mixed-integer optimization in which variables take values from a set <i> S</i>, is introduced and studied in the last part of this dissertation. We generalize two traditional optimization algorithms to <i>S</i>-optimization: the chance-constrained algorithm of Calafiore and Campi, and the sampling algorithm of Clarkson. Interestingly, the complexities of the generalized algorithms are dependent upon the same Helly numbers studied elsewhere in this dissertation.</p>
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