Analyse mathematique III: Fonctions analytiques, by Roger Godement

By Roger Godement

Ce vol. III disclose l. a. th?orie classique de Cauchy dans un esprit orient? bien davantage vers ses innombrables utilisations que vers une th?orie plus ou moins compl?te des fonctions analytiques. On montre ensuite remark les int?grales curvilignes ? los angeles Cauchy se g?n?ralisent ? un nombre quelconque de variables r?elles (formes diff?rentielles, formules de variety Stokes). Les bases de l. a. th?orie des vari?t?s sont ensuite expos?es, principalement pour fournir au lecteur le langage "canonique" et quelques th?or?mes importants (changement de variables dans les int?grales, ?quations diff?rentielles). Un dernier chapitre montre touch upon peut utiliser ces th?ories pour construire l. a. floor de Riemann compacte d'une fonction alg?brique, sujet rarement trait? dans los angeles litt?rature non sp?cialis?e bien que n'?xigeant que des strategies ?l?mentaires. Un quantity IV exposera, outre,l'int?grale de Lebesgue, un bloc de math?matiques sp?cialis?es vers lequel convergera tout le contenu des volumes pr?c?dents: s?ries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, th?orie classique des fonctions modulaires et los angeles model moderne utilisant l. a. constitution de groupe de Lie de SL(2,R).

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8. Let (Pk)kEN be a decreasing (with respect to inclusion) sequence of parallelepipeds in Rn. Show (by applying the result of Exercise 7) that the set P = n { Pk : k E N} 47 CONVEX POLYHEDRA is a parallelepiped in R n. This statement is due to Gruber. Let (LkhEN be an increasing (with respect to inclusion) sequence of parallelepipeds in Rn such that the union of all these parallelepipeds is a bounded subset of Rn. Show that the set L = cl(U{Lk : kEN}) is a parallelepiped in R n. 9. Let M be a convex polyhedron in the space Rn which admits a decomposition into finitely many centrally symmetric convex polyhedra.

X, Y, Y, ... , Y) = Vn-k,k where n- k copies of X and k copies of Y participate. In particular, it can easily be observed that Vn-k,k(X, Y) = Vk,n-k(Y, X). 28 BRUNN-MINKOWSKI INEQUALITY In the sequel, only the properties of v1,n-l will be essentially utilized. Let us formulate and prove the next auxiliary statement which will be useful below. Lemma 4. Let M and P be two n-dimensional convex polyhedra in the space Rn, with equal numbers of (n- I)-dimensional faces. Denote by the injective family of all (n- I)-dimensional faces of M and by the injective family of all ( n- 1 )-dimensional faces of P.

Especially, we are interested in the Minkowski classical theorem on the existence and uniqueness of a convex polyhedron with given directions of its exterior normals and given ( n- 1)-dimensional volumes of its (n- 1)-dimensional faces (briefly, (n- 1)-faces). We shall present below the precise formulation of the above-mentioned Minkowski theorem. For this purpose, we begin with some simple statements about exterior normals of a given convex polyhedron. Let M be a convex polyhedron in the space Rn where n > 2.

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