Convex Optimization Techniques for Geometric Covering Problems

The present thesis is a commencement of a generalization of covering results in specific settings such as the Euclidean space or the sphere to arbitrary compact metric spaces. In particular we consider coverings of compact metric spaces $(Xd)$ by balls of radius $r$. We are interested in the minimum number of such balls needed to cover $X$ denoted by $\Ncal(Xr)$. For finite $X$ this problem coincides with an instance of the combinatorial \textsc{set cover} problem which is $\mathrm{NP}$-complete. We illustrate approximation techniques based on the moment method of Lasserre for finite graphs and generalize these techniques to compact metric spaces $X$ to obtain upper and lower bounds for $\Ncal(Xr)$. \\ The upper bounds in this thesis follow from the application of a greedy algorithm on the space $X$. Its approximation quality is obtained by a generalization of the analysis of Chv\'atal's algorithm for the weighted case of \textsc{set cover}. We apply this greedy algorithm to the spherical case $X=S^n$ and retrieve the best non-asymptotic bound of B\or\oczky and Wintsche. Additionally the algorithm can be used to determine coverings of Euclidean space with arbitrary measurable objects having non-empty interior. The quality of these coverings slightly improves a bound of Nasz\'odi. \\ For the lower bounds we develop a sequence of bounds $\Ncal^t(Xr)$ that converge after finitely (say $\alpha\in\N$) many steps: $$\Ncal^1(Xr)\leq \ldots \leq \Ncal^\alpha(Xr)=\Ncal(Xr).$$ The drawback of this sequence is that the bounds $\Ncal^t(Xr)$ are increasingly difficult to compute since they are the objective values of infinite-dimensional conic programs whose number of constraints and dimension of underlying cones grow accordingly to $t$. We show that these programs satisfy strong duality and derive a finite dimensional semidefinite program to approximate $\Ncal^2(S^2r)$ to arbitrary precision. Our results rely in part on the moment methods developed by de Laat a