O intuito da presente dissertação é o estudo do chamado problema de Kakeya. Mais precisamente, depois introduzir todos os pré-requisitos necessários, demostraremos a conjectura de Kakeya no caso de R2, destacando que o caso geral dessa conjectura, que afirma que cada conjunto de Kakeya de Rn possui dimensão de Hausdorff igual a n, é ainda um problema aberto. / In this dissertation we study the so called Kakeya problem. More precisely, after introducing all the necessary prerequisites, our main goal will be to give a proof of the the Kakeya conjecture in the case of R2, whose general case, stating that every Kakeya set in Rn has Hausdorff dimension equal to n, is still an open problem.

O intuito da presente dissertação é o estudo do chamado problema de Kakeya. Mais precisamente, depois introduzir todos os pré-requisitos necessários, demostraremos a conjectura de Kakeya no caso de R2, destacando que o caso geral dessa conjectura, que afirma que cada conjunto de Kakeya de Rn possui dimensão de Hausdorff igual a n, é ainda um problema aberto. / In this dissertation we study the so called Kakeya problem. More precisely, after introducing all the necessary prerequisites, our main goal will be to give a proof of the the Kakeya conjecture in the case of R2, whose general case, stating that every Kakeya set in Rn has Hausdorff dimension equal to n, is still an open problem.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

ANDRADE, Nicolas Alcântara de. Analiticidade de funções diferenciáveis em quase todo ponto. 2015. 35 f. Dissertação (Mestrado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-09-08T16:45:17Z No. of bitstreams: 1 2013_dis_naandrade.pdf: 869687 bytes, checksum: 0a5b65c36f0e2f52a389f936c5e2b61c (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-09-09T11:47:20Z (GMT) No. of bitstreams: 1 2013_dis_naandrade.pdf: 869687 bytes, checksum: 0a5b65c36f0e2f52a389f936c5e2b61c (MD5) / Made available in DSpace on 2015-09-09T11:47:20Z (GMT). No. of bitstreams: 1 2013_dis_naandrade.pdf: 869687 bytes, checksum: 0a5b65c36f0e2f52a389f936c5e2b61c (MD5) Previous issue date: 2015 / This work is based on the article Analyticity Of Almost Everywhere Differentiable Functions, it will develop a partitioning lemma for superadditive set functions which will lead to a simple alternative proof of Besicovitch’s theorems. / Esse trabalho é baseado no artigo Analyticity Of Almost Everywhere Differentiable Functions, nele desenvolveremos um lema de partição para funções superaditivas que permitirá uma demonstração alternativa e simples dos teoremas de Besicovitch.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esse trabalho à baseado no artigo Analyticity Of Almost Everywhere Differentiable Functions, nele desenvolveremos um lema de partiÃÃo para funÃÃes superaditivas que permitirà uma demonstraÃÃo alternativa e simples dos teoremas de Besicovitch. / This work is based on the article Analyticity Of Almost Everywhere Differentiable Functions, it will develop a partitioning lemma for superadditive set functions which will lead to a simple alternative proof of Besicovitchâs theorems .

In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons. We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods. We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.

In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons. We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods. We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.

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V této práci se zabýváme r·znými pokrývacími větami a jejich ap- likacemi. Kromě klasických pokrývacích vět (Vitaliova, Besicovitchova a Whitney- ova věta) zde uvádíme i některá jejich zobecnění a další pokrývací věty. Tyto věty pak používáme v d·kazech dalších vět, některé jsou typickými aplikacemi pokrý- vacích vět jako například Lebesgueova věta o derivování, slabý typ (1,1) maximál- ního operátoru nebo Calderónovo-Zygmundovo lemma, v jejichž d·kazech hrají pokrývací věty klíčovou roli. Dále se zabýváme nerovnostmi mezi operátory, po- mocí pokrývacích vět dokazujeme vztahy mezi Hardyovým-Littlewoodovým max- imálním operátorem, maximálním singulárním integrálním operátorem a ostrým maximálním operátorem. 1