| In this thesis, we consider a selection of extremal problems which arise in mathematical analysis. The problems themselves come from a variety of fields of mathematics including the theory of functional equations, discrete geometry and harmonic analysis.;In the second chapter, we study the Cauchy-Pexider functional equation f(x)g(y) = h(x+y), and some variations, where x,y are restricted to lie on a submanifold of a real Euclidean space. Our main result gives sufficient geometric conditions so that when this equation is satisfied for x,y lying on some hypersurface the solutions are essentially unique and extend to exponentially affine functions on the entire Euclidean space.;In the third chapter, we briefly describe the Erdős distance problem which underlies much of the motivation for the problems studied in the third and fourth chapters. The remainder of the chapter is devoted to improving the lower bound for distinct distance subsets of finite sets in the plane, sphere and hyperbolic plane. More precisely, we show that given a finite subset P of the Euclidean plane, the sphere or the hyperbolic plane, there exists a subset P of cardinality Q [formula omitted] such that all the distances between pairs of points in Q (measured in the standard metric for each space) are distinct.;In the fourth chapter, we study distinct distances on algebraic curves embedded in real Euclidean space. We show that N points on a real algebraic curve of degree n in integers d always determine [greater than or equivalent to] n,d N1+¼ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of N points which determine [less than or equivalent to] N distinct distances. Our method may be applied to other quantities of interest to obtain analogous exponent gaps. An important step in the proof involves understanding the structural rigidity of certain frameworks on curves.;In the fifth chapter, we characterize the near-extremizers of Young's convolution inequality for discrete, torsion-free groups. Using our result, we also characterize the near-extremizers of the Hausdorff-Young inequality for integersd.. |
