Random covering in high dimension by a union of scaled convex sets

This PhD thesis is devoted to random covering theory; we study the covering property of a set by a union of randomly placed sets, and focus mainly on the condition for almost sure coverage of every point of the set. A. Dvoretzky initiated this direction of research by proving that covering every fixed point with probability 1 does not necessarily imply that every point is covered with probability 1 when the set to be covered is uncountable, by giving an example where covering every point in a unit circumference circle almost surely does not imply covering the whole circle [6]. Since then, to study this phenomenon, several settings have been proposed; we concentrate on two of these, the Dvoretzky problem and the Mandelbrot problem. For the Dvoretzky problem, let $calC$ be a convex set and let ${ v_n }$ be a sequence of volumes of scaled copies of $calC$ that are placed uniformly on the d-dimensional torus. We find a necessary condition and also a sufficient condition for the union of the sets to cover a fixed k-dimensional hyperplane, $k > 0$. Furthermore, a necessary and sufficient condition is also obtained for the special case when $k=1$. For the Mandelbrot problem, let C be a convex set with volume 1 in $R^d$, and let each point $(x, z)$, where $x in R^d$ and $z in R^+$ be associated with a convex set $x + z C$. Let $Phi$ be a Poisson point process in $R^d imes R^+$ with intensity $lambdaotimesmu$, where $lambda$ is a Lebesgue measure and $mu$ is a $sigma$-finite measure. We give a necessary condition and also a sufficient condition on $mu$ for the union of all convex sets associated with points in $Phi$ to cover any k-dimensional hyperplane in $R^d$. Furthermore, a necessary and sufficient condition is also obtained for the special case when $k=1$. We also consider covering a more general set. In particular, we derive a necessary condition and also a sufficient condition for covering a Cantor set and its generalized version in the one-dimensional Mandelbrot problem setting.Ph. D.Includes bibliographical referencesIncludes vitaby Tuan H. Nguye