Equivariant absolute extensor property on hyperspaces of convex sets

Let G be a compact group acting on a Banach space L by means of linear isometries. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.Comment: 8 page

Local compactness in right bounded asymmetric normed spaces

[EN] We characterize the ¿nite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, such as compactness and strong compactness. In contrast with some results found in the ex-isting literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are ¿nite dimen-sional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open ques-tion on the topology of ¿nite dimensional asymmetric normed spaces. In the positive direction, we will prove that a ¿nite dimensional asym-metric normed space is strongly locally compact if and only if it is right bounded.The first author has been supported by Conacyt grant 252849 (Mexico) and by PAPIIT grant IA104816 (UNAM, Mexico). The second author has been supported by Ministerio de Economia y Competitividad (Spain) (project MTM2016-77054-C2-1-P)Jonard Pérez, N.; Sánchez Pérez, EA. (2018). Local compactness in right bounded asymmetric normed spaces. Quaestiones Mathematicae. 41(4):549-563. https://doi.org/10.2989/16073606.2017.1391351S54956341

On the topology of some hyperspaces of convex bodies associated to tensor norms

For every tuple $d_1,\dots, d_l\geq 2,$ let $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}$ denote the tensor product of $\mathbb{R}^{d_i},$ $i=1,\dots,l.$ Let us denote by $\mathcal{B}(d)$ the hyperspace of centrally symmetric convex bodies in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ endowed with the Hausdorff distance, and by $\mathcal{B}_\otimes(d_1,\dots,d_l)$ the subset of $\mathcal{B}(d)$ consisting of the convex bodies that are closed unit balls of reasonable crossnorms on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ It is known that $\mathcal{B}_\otimes(d_1,\dots,d_l)$ is a closed, contractible and locally compact subset of $\mathcal{B}(d).$ The hyperspace $\mathcal{B}_\otimes(d_1,\dots,d_l)$ is called the space of tensorial bodies. In this work we determine the homeomorphism type of $\mathcal{B}_\otimes(d_1,\dots,d_l).$ We show that even if $\mathcal{B}_\otimes(d_1,\dots,d_l)$ is not closed with respect to the Minkowski sum, it is an absolute retract homeomorphic to $\mathcal{Q}\times\mathbb{R}^p,$ where $\mathcal{Q}$ is the Hilbert cube and $p=\frac{d_1(d_1+1)+\cdots+d_l(d_l+1)}{2}.$ Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to $\mathcal{B}_\otimes(d_1,\dots,d_l)$ is examined.Comment: 28 pages. Among others, in this version we added an illustrative figure for the proof of Lemma 2.6. A gap on the selection of $r$ in the proof of lemma 2.6 was corrected. We provide a new sentence for Proposition 5.2. This new statement improves the resul

A Simple Proposal for Including Designer Preferences in Multi-Objective Optimization Problems

[EN] Including designer preferences in every phase of the resolution of a multi-objective optimization problem is a fundamental issue to achieve a good quality in the final solution. To consider preferences, the proposal of this paper is based on the definition of what we call a preference basis that shows the preferred optimization directions in the objective space. Associated to this preference basis a new basis in the objective space-dominance basis-is computed. With this new basis the meaning of dominance is reinterpreted to include the designer's preferences. In this paper, we show the effect of changing the geometric properties of the underlying structure of the Euclidean objective space by including preferences. This way of incorporating preferences is very simple and can be used in two ways: by redefining the optimization problem and/or in the decision-making phase. The approach can be used with any multi-objective optimization algorithm. An advantage of including preferences in the optimization process is that the solutions obtained are focused on the region of interest to the designer and the number of solutions is reduced, which facilitates the interpretation and analysis of the results. The article shows an example of the use of the preference basis and its associated dominance basis in the reformulation of the optimization problem, as well as in the decision-making phase.This work has been supported by the Ministerio de Ciencia, Innovacion y Universidades, Spain, under Grant RTI2018-096904-B-I00.Blasco, X.; Reynoso Meza, G.; Sánchez Pérez, EA.; Sánchez Pérez, JV.; Jonard Pérez, N. (2021). A Simple Proposal for Including Designer Preferences in Multi-Objective Optimization Problems. Mathematics. 9(9):1-19. https://doi.org/10.3390/math90909911199