| The Ham Sandwich Theorem states that under very general conditions, any n finite measures on Rn can be bisected by a single hyperplane. This thesis generalizes this theorem in two directions, utilizing techniques from algebraic and geometric topology.;In one direction, algebraic analogues are obtained via equivariant methods. Viewing the Ham Sandwich theorem as stating that any n real measures on R n can be "Z2-equipartitioned" by a single pair of half-spaces, generalizations are obtained for the cyclic groups Zm by showing that any n complex-valued measures on Cn can be "Z m-equipartitioned" by a single collection of regular sectors, and for the finite subgroups G of S 3 by showing that any n quaternion-valued measures on Hn can be "G-equipartitioned" by a single collection of fundamental G-regions. Real equipartition results are obtained as corollaries, among them the even- dimensional Ham Sandwich Theorem and that any n measures on R 2n can be (a) trisected by a regular 3-fan and (b) bisected by a pair of orthogonal hyperplanes.;In another direction, we are motivated by (b) to seek the maximum number o( m,n) of pairwise orthogonal hyperplanes which bisect any given m measures on Rn, m≤ n. An "orthogonal" ham sandwich theorem is found in the form of a lower bound for o(m,n), constructed using orthonormal vector fields on real projective space. As optimal cases, we obtain o(2, 4) = 3 and o(2, 8) = 7, corresponding to the parallelizability of R P3 and RP7.;The thesis concludes by unifying these two directions of generalization via orthogonal versions of the algebraic ham sandwich theorems. These are found by replacing Z2-equivariant orthonormal vector fields on real spheres with Zm-equivariant complex-orthonormal vector fields on complex spheres, and similarly in the quaternionic cases. Real equipartition statements follow, among them that any n measures on R 4n can be trisected by a pair orthogonal regular 3-fans, with the case n = 1 being optimal. |
