In this dissertation we define a generalization of Kakeya sets in certainmetric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesguemeasure containing a segment of length one in every direction. A famousconjecture, known as Kakeya conjecture, states that the Hausdorff dimension ofany Kakeya set should equal the dimension of the space. It was proved only inthe plane, whereas in higher dimensions both geometric and arithmeticcombinatorial methods were used to obtain partial results. In the first part ofthe thesis we define generalized Kakeya sets in metric spaces satisfyingcertain axioms. These allow us to prove some lower bounds for the Hausdorffdimension of generalized Kakeya sets using two methods introduced in theEuclidean context by Bourgain and Wolff. With this abstract setup we can dealwith many special cases in a unified way, recovering some known results andproving new ones. In the second part we present various applications. Werecover some of the known estimates for the classical Kakeya and Nikodym setsand for curved Kakeya sets. Moreover, we prove lower bounds for the dimensionof sets containing a segment in a line through every point of a hyperplane andof an (n-1)-rectifiable set. We then show dimension estimates for Furstenbergtype sets (already known in the plane) and for the classical Kakeya sets withrespect to a metric that is homogeneous under non-isotropic dilations and inwhich balls are rectangular boxes with sides parallel to the coordinate axis.Finally, we prove lower bounds for the classical bounded Kakeya sets and anatural modification of them in Carnot groups of step two whose second layerhas dimension one, such as the Heisenberg group. On the other hand, if thedimension is bigger than one we show that we cannot use this approach.
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