# RESEARCH INTERESTS

My general research interest is in Geometry of infinite dimensional Banachspaces. This research falls within the field of Functional Analysis. My current research plans are concentrated within the following general areas:

• Renormings of Banachspaces. A central problem within the Banach space theory is that of knowing if a given space has or has not norms (i.e. admits or not renormings) with good smoothness and/or convexity properties. This is important because isomorphic properties can be deduced from this information.
• Norm attaining mappings.The structure of the so-called Bishop-Phelps set on a given Banachspace X (i.e, the set of norm attaining functionalson X) is of interest in my research. Within this subject, there are very nice,  interesting problems we are attempting. Some of them are related to the study of those Banachspaces where a «generalization» of the Bishop-Phelps Theorem holds, namely, the density of the set of norm attaining operators or multilinear mappings or polynomials.
• The metric space (H(X),h) of all bounded, closed and convex sets of a Banach space X with the Hausdorff metric. One of the most important reasons why we are interested in this space is because many properties within the theory of Banach spaces can be reformulated in the following way: «every subset of H(X) satisfies a given condition». This is the case, for instance, of the well-known Radon Nikodym property (geometrical version). Thus, where X does not possess such property, it is interesting to study the topological aspects of the subset of H(X) which is still satisfying the given condition.
• Smooth functions and smooth bump functions. The study of the existence of smooth (Gâteaux,Fréchet, etc…) bump functions and its properties on certain Banach spaces plays a central role within the geometry of Banachspaces. We are mainly interested in: (1) the range and the size of the set of gradients of smooth bump functions and more general smooth functions; (2) properties of approximation such as approximation of Lipschitz functions by \$C^1\$-smooth and Lipschitz functions; (3) properties of extensions such as \$C^1\$-smooth extensions of functions defined on a closed subset of a Banach space X.
• Banach-Finsler Manifolds. The study of smoothness properties on Banach-Finsler manifolds such as smooth approximation of functions, non-smooth analysis and Hamilton-Jacobi equations on Banach-Finsler manifolds.

# PUBLICATIONS

Normal and starlike tilings in separable Banach spaces.

R. Deville, M. Jiménez-Sevilla, Journal of Mathematical Analysis and Applications, 500 (2021) no. 2, 125116.

A class of Hamilton-Jacobi equations on Banach-Finsler manifolds.

J. A. Jaramillo, M. Jiménez-Sevilla, J. L. Ródenas and L. Sánchez-González,
Nonlinear Analysis: Theory, Methods and Applications, 113(2015), 159–179.

J. A. Jaramillo, M. Jiménez-Sevilla and Luis Sánchez-González,
Proceedings of the American Math. Soc. 142 (2014) no. 3, 1075–1087.

On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces.

M. Jiménez-Sevilla and Luis Sánchez-González,
Math. Ann. 355 (2013), 1201-1219.

On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler Manifolds.

M. Jiménez-Sevilla and Luis Sánchez-González.
Nonlinear Analysis: Theory, Methods & Applications,74 (2011), 3487-3500.

M. Jiménez-Sevilla and Luis Sánchez-González, Journal of Mathematical Analysis and Applications, 378 (2011), 173-183.

M. Jiménez-Sevilla and Luis Sánchez-González,
Journal of Mathematical Analysis and Applications, 365 (2010), 315-319.

M. Jiménez-Sevilla
Journal of Mathematical Analysis and Applications 348 (2008), n. 2, 573-580.

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds.

D. Azagra, M. Jiménez-Sevilla and F. Macià,
Cal. Var. Partial Differential Equations 33 (2008), n. 2, 133-167.

Approximations by smooth functions with no critical points on separable Banach spaces.

D. Azagra and M. Jiménez-Sevilla,
Journal of Functional Analysis 242 (1) (2007), 1-36.

Exact filling of figures with the derivatives of smooth mappings between Banach spaces.

D. Azagra, M.Fabian, M. Jiménez-Sevilla
Canadian Mathematical Bulletin 48, no. 4 (2005), 481-499.

Intersection of closed balls and geometry of Banach spaces (Survey).

Antonio S. Granero, M. Jiménez-Sevilla, Jose Pedro Moreno
Extracta Mathematicae  19 (num. 2) (2004), 55-92.

On the Kunen-Shelah properties Banach spaces.

A. S. Granero, Mar Jiménez-Sevilla, J. P. Moreno, A. Montesinos y A. Plichko
Studia Mathematica 157 (no. 2), (2003), 97-120.

On the Range of the derivatives of a smooth function between Banach spaces.

D. Azagra, R.Deville, M. Jiménez-Sevilla
Math. Proc. Cambdrige Philosophical Society 134 (num. 1), (2003), 163-185.

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space.

D. Azagra, M. Jiménez-Sevilla
Bull. Soc. Math. Franc. 130, (2002), 337-347.

Geometrical and topological properties of bumps and starlike bodies in Banach spaces.

D. Azagra, M. Jiménez-Sevilla
Extracta Mathematicae 17(2), (2002), 151-2002.

Antiproximinal norms in Banach spaces

J. M. Borwein, M. Jiménez-Sevilla, J. P. Moreno
Journal of Approximation Theory 114, (2002), 57-69.

On w-independence and the Kunen-Shelah property

A. S. Granero, M. Jiménez-Sevilla, J. P. Moreno
Proc. Edinburgh Math. Soc. 45, (2002), 391-395.

Complementation and embbedings of co(I) in Banach spaces

S. A. Argiros, J. F. Castillo, A. S. Granero, M. Jiménez-Sevilla y J. P. Moreno
Proceedings of the London Mathematical Society 85(3), (2002), 742-768.

The failure of Rolle’s theorem in infinite dimensional Banach spaces

D.Azagra, M. Jiménez-Sevilla
Journal of Functional Analysis, 182(2001), 207-226.

A constant of porosity for convex bodies

M. Jiménez-Sevilla, J.P. Moreno
Illinois Journal of Mathematics 45 (2001), 1061-1071.

A note on porosity and Mazur intersection property

M. Jiménez-Sevilla, J.P. Moreno
Mathematika 47 (2000), 267-272.

On the non-separable subspaces J(h) and C[1,h]

Antonio S. Granero, M. Jiménez-Sevilla, J.P. Moreno
Mathematishe Nachrichten 221 (2001), 75-85.

Mazur intersection properties and differentiability of convex functions in Banach spaces

P. S. Georgiev, Antonio S. Granero, M. Jiménez-Sevilla, J.P. Moreno
J. London Math. Soc. (2) 61 (2000) no. 2, 531-542

Sequential continuity in the ball topology of a Banach space

Antonio S. Granero, M. Jiménez-Sevilla, J.P. Moreno,
Indagationes Mathematicae 10 (3) (1999), 423-435

Geometry of Banach spaces with property beta

Antonio S. Granero, M. Jiménez-Sevilla, J.P. Moreno,
Israel Journal of Mathematics 111 (1999), 263-273

A note on norm attaining functionals

M. Jiménez-Sevilla, J.P. Moreno
Proc. Am. Math. Soc. 126 (1998), 1989-1997

Convex sets on Banach spaces and a problem of Rolewicz

Antonio S. Granero, M. Jiménez-Sevilla, J. P. Moreno
Studia Mathematica 129 (1) (1998), 19-29

Norm attaining multilinear forms and polynomials on predual of Lorentz sequence spaces

Rafael Payá, M. Jiménez-Sevilla
Studia Mathematica 127 (2) (1998), 99-112.

Renorming Banach spaces with the Mazur intersections Property

M. Jiménez-Sevilla, J.P. Moreno
Journal Functional Analysis 144 (2) (1997), 486-504

On denseness of certain norms in Banach spaces

M. Jiménez-Sevilla, J.P. Moreno
Bull. Austral. Math. Soc. 54 (1996), 183-196.

The Mazur intersection property and Asplund spaces

M. Jiménez-Sevilla, J.P. Moreno
C. R. Acad.Sci. Paris, t. 321, Série I, p. 1219-1223, (1995).

# TESIS (PhD DISSERTATION)

Tesina realizada: Funciones convexas en espacios de Banach