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Julián Eduardo Haddadhttp://lattes.cnpq.br/8883222618253604Monika LudwigFabian MußnigCarlos Hugo Jimenez GomezRafael Villa CaroVitor Balestro Dias da Silvahttp://lattes.cnpq.br/5066259198157701Fernanda Helen Moreira Baeta2024-05-09T15:44:35Z2024-05-09T15:44:35Z2024-02-21http://hdl.handle.net/1843/68148This thesis consists in two separate parts, each studying a different problem in the theory of convex bodies. The first part deals with isotropic measures, more specifically, the problem of describing explicitly the weights in the decomposition of the identity for a convex body in John position. We do this for the John position, that is, when the n-dimensional unit Euclidean ball B^n, is the ellipsoid with maximum volume inside K, and for the positive John position with respect to the convex body L, that is, when L ⊆ K and L has maximal volume among all images TL in K, where T is a positive-definite matrix. We also do this for functional ellipsoids in the sense defined by Ivanov and Naszódi [30]. We consider proper log-concave functions h : R^n → R (log-concave and upper semicontinuous functions that has finite positive integral). By [30], for every s > 0 there is (and is unique in the set of proper log-concave functions) one log-concave function with the largest integral under the condition that it is pointwise less than or equal to h^{1/s}. This function is called John s-function of h. Again, by [30] there exists a characterization of this function similar to the one given by John in his fundamental theorem. The second part studies the problem of characterizing upper semicontinuous valuations. Denote by Conv_pac(R; R) the space of finite-valued, convex functions on R that are piecewise affine outside of a compact set. A functional Z : Conv_pac(R; R) → R is called a valuation if Z(u ∨ v) + Z(u ∧ v) = Z(u) + Z(v) for all u, v ∈ Conv_pac(R; R) such that u ∨ v, u ∧ v ∈ Conv_pac(R; R). Here, u ∨ v and u ∧ v denote the pointwise maximum and minimum of u, v ∈ Conv_pac(R; R), respectively. A classification of upper semicontinuous, translation invariant valuations and unchanged by the addition of piecewise affine functions on the space Conv_pac(R; R) is established.Esta tese consiste em duas partes distintas, cada uma estudando um problema diferente na teoria dos corpos convexos. A primeira parte trata das medidas isotrópicas, mais especificamente, do problema de descrever explicitamente os pesos na decomposição da identidade para um corpo convexo na posição de John. Fazemos isso para a posição de John, ou seja, quando a bola Euclidiana unitária n-dimensional Bn, é o elipsóide com volume máximo dentro de K, e para a posição positiva de John em relação ao corpo convexo L, ou seja, quando L ¦ K e L tem volume máximo dentre todas as imagens TL em K, onde T é uma matriz definida-positiva. Também fazemos isso para elipsóides funcionais no sentido definido por Ivanov e Naszódi [30]. Consideramos funções log-côncavas próprias h : Rn → R (funções log-côncavas e semicontínuas superiores que possuem integral positiva finita). Por [30], para cada s > 0 existe (e é única no conjunto de funções log-côncavas próprias) uma função log-côncava com a maior integral sob a condição de que esta seja pontualmente menor ou igual a h1/s. Essa função é chamada s-função de John de h. Novamente, por [30], existe uma caracterização dessa função semelhante àquela dada por John em seu teorema fundamental. A segunda parte estuda o problema de caracterização de valuações semicontínuas superiores. Denote por Convpac(R;R) o espaço de funções convexas de valor finito em R que são afins por partes fora de uma conjunto compacto. Um funcional Z : Convpac(R;R) → R é chamado uma valuação se Z(u ( v) + Z(u ' v) = Z(u) + Z(v) para todo u, v ∈ Convpac(R;R) tal que u ( v, u ' v ∈ Convpac(R;R). Aqui, u ( v e u ' v denotam as funções máximo e mínimo pontuais de u, v ∈ Convpac(R;R), respectivamente. Uma classificação de valuações semicontínuas superiores, invariantes por translação e inalterada por adição de funções afins por partes no espaço Convpac(R;R) é estabelecida.CNPq - Conselho Nacional de Desenvolvimento Científico e TecnológicoCAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAhttp://creativecommons.org/licenses/by-nc-nd/3.0/pt/info:eu-repo/semantics/openAccessMatemática - TesesCorpos convexos - TesesFunções convexas – TesesTeoria da valorização – TesesConvex bodyJohn positionLöwner positionDecomposition of the identityIsotropic measuresLog-concave functionsFunctional ellipsoidsValuations on the space of convex functionsTwo problems on convex geometry: isotropic measures and classification in valuation theoryinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALTwo problems on convex geometry.pdfTwo problems on convex geometry.pdfapplication/pdf1519112https://repositorio.ufmg.br/bitstream/1843/68148/2/Two%20problems%20on%20convex%20geometry.pdf6d77a37d9beaed664ba89f5f2ae1089fMD52CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8811https://repositorio.ufmg.br/bitstream/1843/68148/3/license_rdfcfd6801dba008cb6adbd9838b81582abMD53LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/68148/4/license.txtcda590c95a0b51b4d15f60c9642ca272MD541843/681482024-05-09 12:44:36.484oai:repositorio.ufmg.br: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ório InstitucionalPUBhttps://repositorio.ufmg.br/oaiopendoar:2024-05-09T15:44:36Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false
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