Two Topics in Geometry: Minimizing Absolute Gaussian Curvature Locally and Solution Path of the Slab Support Vector Machine

In this thesis we deal with two geometric optimization problems: Minimizing Absolute Gaussian Curvatue Locally in Meshes and the Solution path of the Slab Support Vector Machine. In the first part, we show that given a polygon C in R it is in general algebraically hard to find a point in R3\C at which the absolute Gaussian curvature with respect to C is minimized. By algebraically hard we mean that in general an optimal solution is not constructible, i.e., there exist no finite sequence of expressions starting with rational numbers, where each expression is either the sum, difference, product, quotient or k’th root of preceding expressions and the last expressions give the coordinates of an optimal solution. This essentially only leaves to approximate such an optimal point. We provide an approximation scheme for the minimum value of the absolute curvature. In the second part, we design an algorithm to compute the solution path of the slab support vector machine. More specifically, given a set of points in a Hilbert space that can be separated from the origin, the slab support vector machine (slab SVM) is an optimization problem that aims at finding a slab (two parallel hyperplanes whose distance is essentially fixed) that encloses the points and is maximally separated from the origin. Special cases of the slab SVM include the smallest enclosing ball problem and an interpolation problem that was used (like the slab SVM itself) in surface reconstruction with radial basis functions. Here we show that the path of solutions of the slab SVM, i.e., the solution parameterized by the slab width is piecewise linear. We also give an algorithm to compute the whole solution path.