### Browsing by Author "Jiménez Sevilla, María del Mar"

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Publication A class of Hamilton-Jacobi equations on Banach-Finsler manifolds(Pergamon-Elsevier, 2015) Jaramillo Aguado, Jesús Ángel; Jiménez Sevilla, María del Mar; Rodenas Pedregosa, J.L.; Sánchez González, L.The concept of subdifferentiability is studied in the context of C-1 Finsler manifolds (modeled on a Banach space with a Lipschitz C-1 bump function). A class of Hamilton-Jacobi equations defined on C-1 Finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions are obtained.Publication A constant of porosity for convex bodies(University of Illinois, 2001) Jiménez Sevilla, María del Mar; Moreno, José PedroIt was proved recently that a Banach space fails the Mazur intersection property if and only if the family of all closed, convex and bounded subsets which are intersections of balls is uniformly very porous. This paper deals with the geometrical implications of this result. It is shown that every equivalent norm on the space can be associated in a natural way with a constant of porosity, whose interplay with the geometry of the space is then investigated. Among other things, we prove that this constant is closely related to the set of ε-differentiability points of the space and the set of r-denting points of the dual. We also obtain estimates for this constant in several classical spaces.Publication A note on norm attaining functionals(American Mathematical Society, 1998-07) Jiménez Sevilla, María del Mar; Moreno, José PedroWe are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space X is reflexive if and only if it satisfies any of the following properties: (i) X admits a norm |· | with the Mazur Intersection Property and the set N A|· | of all norm attaining functionals of X* contains an open set, (ii) the set N A|· | 1 of all norm one elements of N A|· | contains a (relative) weak* open set of the unit sphere, (iii) X* has C* PCP and N A|· | 1 contains a (relative) weak open set of the unit sphere, (iv) X is WCG, X* has CPCP and N A|· | 1 contains a (relative) weak open set of the unit sphere. Finally, if X is separable, then X is reflexive if and only if N A|· | 1 contains a (relative) weak open set of the unit sphere.Publication A note on porosity and the Mazur intersection property(Cambridge Univ. Press, 2000-12) Jiménez Sevilla, María del Mar; Moreno, José PedroLet M be the collection of all intersections of balls, considered as a subset of the hyperspace H of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. We prove that M is uniformly very porous if and only if the space fails the Mazur intersection property.Publication A note on the range of the derivatives of analytic approximations of uniformly continuous functions on co(Elsevier, 2008) Jiménez Sevilla, María del MarThis paper is a contribution to the body of results concerning the size of the set of derivatives of differentiable functions on a Banach space. The results so far have consisted of examples of highly differentiable bump functions (or functions approximating a given continuous function) whose set of derivatives either is surprisingly small or has a given shape. The paper under review treats the case of analytic smoothness. The main result states that every uniformly continuous function on c0 (more generally on a space with property (K)) can be approximated by a real-analytic function whose set of derivatives is contained in T p>0 lp. This is a significant step forward, as analytic functions are substantially harder to deal with than C1 smooth ones. Indeed, a local perturbation of an analytic function necessarily changes the values of the function everywhere. Also of particular value is the quite precise and elegant description of the set of derivatives of the approximating function.Publication Antiproximinal norms in Banach spaces(Academic Press-Elsevier Science, 2002-01) Borwein, Jonathan M.; Jiménez Sevilla, María del Mar; Moreno, José PedroWe prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal norms in Banach spaces with the Convex Point of Continuity Property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed.Publication Approximation by smooth functions with no critical points on separable Banach spaces(Elsevier, 2007-06-01) Azagra Rueda, Daniel; Jiménez Sevilla, María del MarWe characterize the class of separable Banach spaces X such that for every continuous function f : X -> Rand for every continuous function epsilon : X -> (0, +infinity) there exists a C-1 smooth function g: X -> R for which vertical bar f(x) - g(x)vertical bar <= epsilon(x) and g'(x) not equal 0 for all x is an element of X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X*. We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class C-p, for p = 1, 2,..., +infinity. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces l(p)(N) and L-p(R-n). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.Publication Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions(American Mathematical Society, 2014-03) Jaramillo Aguado, Jesús Ángel; Jiménez Sevilla, María del Mar; Sánchez González, L.In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete C-k Finsler manifold M is determined by the normed algebra C-b(k)(M) of all real-valued, bounded and C-k smooth functions with bounded derivative defined on M. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete C-k Finsler manifold M is determined by the algebra C-b(k)(M); (ii) the weak Finsler structure of a separable and complete C-k Finsler manifold M modeled on a Banach space with a Lipschitz and C-k smooth bump function is determined by the algebra C-b(k)(M); (iii) the weak Finsler structure of a C-1 uniformly bumpable and complete C-1 Finsler manifold M modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra C-b(1)(M); and (iv) the isometric structure of a WCG Banach space X with a C-1 smooth bump function is determined by the algebra C-b(1)(X).Publication Complementation and embeddings of c0(I) in Banach spaces(London Mathematical Society, 2002-11) Argyros, Spiros A.; Castillo, Jesús F.; Granero, A. S.; Jiménez Sevilla, María del Mar; Moreno, José PedroPublication Convex sets in Banach spaces and a problem of Rolewicz(Polish Acad Sciencies Inst Mathematics, 1998) Granero, A. S.; Jiménez Sevilla, María del Mar; Moreno, José PedroLet BX be the set of all closed, convex and bounded subsets of a Banach space X equipped with the Hausdor metric. In the rst part of this work we study the density character of BX and investigate its connections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are concerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).Publication Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces(University of Toronto Press, 2005) Azagra Rueda, Daniel; Fabián, M.; Jiménez Sevilla, María del MarWe establish sufficient conditions on the shape of a set A included in the space Ln s (X; Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a Cn-smooth mapping f : X ¡! Y , with bounded support, and such that f(n)(X) = A, provided that X admits a Cn- smooth bump with bounded n-th derivative and densX = densLn(X; Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n-th derivative of such a mapping. The same holds true for the closure of U, provided that every point in the boundary of U is the end point of a path within U. In the finite-dimensional case, more restrictive conditions are required. We also study the Fr´echet smooth case for mappings from Rn to a separable infinite-dimensional Banach space and the Gˆateaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.Publication Generalized motion of level sets by functions of their curvatures on Riemannian manifolds(Springer, 2008) Azagra Rueda, Daniel; Jiménez Sevilla, María del Mar; Macia Lang, FabricioWe consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function u;M -> R evolve in such a way whenever u solves an equation u (t) + F(Du, D(2) u) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off {Du = 0}, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D(2) u. Finally we give some counterexamples showing that several well known properties of the evolutions in R(n) are no longer true when M has negative sectional curvature.Publication Geometrical and Topological Properties of Bumps and Starlike Bodies in Banach Spaces(Universidad de Extremadura, Departamento de Matemáticas, 2002) Azagra Rueda, Daniel; Jiménez Sevilla, María del MarWhile the topological and geometrical properties of convex bodies in Banach spaces are quite well understood (including their topological and smooth classification), much less is known about the structure of starlike bodies. Starlike bodies are important objects in nonlinear functional analysis as they appear as level sets of $n$-homogeneous polynomials on Banach spaces. Significant progress in the study of starlike bodies has been done in the last years by the efforts of Manuel Cepedello, Robert Deville, Tadeusz Dobrowolski, Marian Fabian and of the authors of the present survey. Its aim is to present these results in a coherent way, emphasizing the connections between infinite-dimensional topology and nonlinear functional analysis (such as the failure of Rolle's and Brouwer's theorems in infinite dimensions), and leading to new characterizations of smoothness properties of Banach spaces. One of the leading ideas of the paper is the use of bump functions as an instrument to study the properties of starlike bodies.\par The paper is divided into eight sections, their headings reflecting the contents and the organization of the paper: 1. Introduction; 2. Classifying starlike bodies; 3. Smooth Lipschitz contractibility of boundaries of starlike bodies in infinite dimensions; 4. The failure of Rolle's and Brouwer's theorems in infinite dimensions; 5. How small can the range of a derivative be? 6. How large does the range of a derivative look like? 8. Geometrical properties of starlike bodies. The failure of James' theorem for starlike bodies.\par Not all the proofs are given in full detail, the authors' emphasis being rather on the ideas lying behind them and on the connections between various properties and notions, avoiding cumbersome details. This survey is a valuable addition to the existing literature and can be used as a guide to this very active area of investigation in nonlinear functional analysis and infinite-dimensional topology.Publication Geometry of Banach spaces with property β(Hebrew University Magnes Press, 1999-12) Granero, A. S.; Jiménez Sevilla, María del Mar; Moreno, José PedroWe prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L 1[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact spacePublication Intersections of closed balls and geometry of Banach spaces(Universidad de Extremadura, Departamento de Matemáticas, 2004) Granero, A. S.; Jiménez Sevilla, María del Mar; Moreno, José PedroIn section 1 we present definitions and basic results concerning the Mazur intersection property (MIP) and some of its related properties as the MIP* . Section 2 is devoted to renorming Banach spaces with MIP and MIP*. Section 3 deals with the connections between MIP, MIP* and differentiability of convex functions. In particular, we will focuss on Asplund and almost Asplund spaces. In Section 4 we discuss the interplay between porosity and MIP. Finally, in section 5 we are concerned with the stability of the (closure of the) sum of convex sets which are intersections of balls and with Mazur spaces.Publication LFC bumps on separable Banach spaces(Elsevier, 2010) Jiménez Sevilla, María del Mar; Sánchez González, LuisIn this note we construct a C∞-smooth, LFC (Locally depending on Finitely many Coordinates) bump function, in every separable Banach space admitting a continuous, LFC bump function.Publication Mazur intersection properties and differentiability of convex functions in Banach spaces(London Mathematical Sociey, 2000-04) Georgiev, P. G.; Granero, A. S.; Jiménez Sevilla, María del Mar; Moreno, José PedroIt is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [script l]1(Γ) and [script l][infty infinity](Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.Publication Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces.(Polish Acad Sciencies Inst Mathematics, 1998) Jiménez Sevilla, María del Mar; Payá Albert, RafaelFor each natural number N, we give an example of a Banach space X such that the set of norm attaining N{linear forms is dense in the space of all continuous N{linear forms on X, but there are continuous (N +1){linear forms on X which cannot be approximated by norm attaining (N+1){linear forms. Actually, X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.Publication Normal and starlike tilings in separable Banach spaces(Elsevier, 2021-03-05) Deville, Robert; Jiménez Sevilla, María del MarIn this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r and R (not depending on the separable Banach space) such that every tile of this tiling is starlike, contains a ball of radius r and is contained in a ball of radius R.Publication On denseness of certain norms in Banach spaces(Cambridge University Press, 1996) Jiménez Sevilla, María del Mar; Moreno, José PedroWe give several results dealing with denseness of certain classes of norms with many vertex points. We prove that, in Banach spaces with the Mazur or the weak* Mazur intersection property, every ball (convex body) can be uniformly approximated by balls (convex bodies) being the closed convex hull of their strongly vertex points. We also prove that given a countable set F, every norm can be uniformly approximated by norms which are locally linear at each point of F.