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Natalia Jonard-Pérez

Publications and source records attributed to Natalia Jonard-Pérez.

15 recordsLinked to original sources
arXiv2026

Decreasing involutions and $\mathbb{Z}_{2}$-means

An $n$-mean on a topological space $X$ is a continuous symmetric map $p\colon X^n\to X$ satisfying $p(x,\ldots,x)=x$ for every $x\in X$. If $X$ is a $G$-space, such an $n$-mean is equivariant if $p(gx_1,\ldots,gx_n)=g p(x_1,\ldots,x_n)$ for every $g\in G$ and $x_1,\ldots,x_n\in X$. For finite $G$, we prove that, in the presence of an equivariant $n$-mean with $n$ a multiple of $\lvert G\rvert$, the $\mathrm{ANE}$, $\mathrm{AE}$, and $\mathrm{AR}$ properties imply their equivariant counterparts. We next consider $\mathbb Z_2$-actions induced by decreasing involutions on topological lattices. We first prove that every fixed point of such an involution on a modular lattice yields an explicit equivariant $2$-mean. Furthermore, we introduce a lattice-theoretic version of the Babylonian iteration used to approximate the geometric mean and prove that, under suitable conditions on the order and its interaction with the topology, this iteration produces an equivariant $2$-mean whenever the lattice admits a $2$-mean compatible with the order. Finally, we apply these results to spaces of compact convex bodies and supercoercive convex functions, where we prove the existence of equivariant $2$-means and show that both spaces are $\mathbb Z_2$-absolute retracts.

math.GN↗
arXiv2025

Equivariant means

An $n$-mean (also called a ''topological social choice rule'') on a topological space $X$ is a continuous function $p:X^n\to X$ satisfying $p(x,\dots, x)=x$ for every $x\in X$ and $p(x_1,\dots, x_n)=p(x_{σ(1)},\dots x_{σ(n)})$ for any permutation $σ$ of $\{1,\dots, n\}$. If, in addition, $X$ is a $G$-space and $p$ is equivariant with respect to the diagonal action of $G$ on $X^n$, we say that $p$ is an equivariant $n$-mean. In this paper, we continue the work initiated by H. Juárez-Anguiano about conditions on a $G$-space $X$, under which the existence of an equivariant $n$-mean guarantees that $X$ is a $G$-AR. We also explore this problem when we remove the symmetry condition on the definition of an $n$-mean.

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arXiv2024

The hyperspace of k-dimensional closed convex sets

For every $n \geq 2$, let $K_k^n$ denote the hyperspace of all $k$-dimensional closed convex subsets of the Euclidean space $R^n$ endowed with the Atouch-Wets topology. Let $ K_{k,b}^n$ be the subset of $K_k^n$ consisting of all $k$-dimensional compact convex subsets. In this paper we explore the topology of $K_k^n$ and $K_{k,b}^n$ and the relation of these hyperspaces with the Grassmann manifold $G_k(n)$. We prove that both $K_k^n$ and $K_{k,b}^n$ are Hilbert cube manifolds with a fiber bundle structure over $G_k(n)$. We also show that the fiber of $K_{k,b}^n$ with respect to this fiber bundle structure is homeomorphic with $\mathbb R^{\frac{k(k+1)+2n}{2}}\times Q$, where $Q$ stands for the Hilbert cube.

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arXiv2023

A topological insight into the polar involution of convex sets

Denote by $\mathcal{K}_0^n$ the family of all closed convex sets $A\subset\mathbb{R}^n$ containing the origin $0\in\mathbb R^n$. For $A\in\mathcal{K}_0^n,$ its polar set is denoted by $A^\circ.$ In this paper, we investigate the topological nature of the polar mapping $A\to A^\circ$ on $(\mathcal{K}_0^n, d_{AW})$, where $d_{AW}$ denotes the Attouch-Wets metric. We prove that $(\mathcal{K}_0^n, d_{AW})$ is homeomorphic to the Hilbert cube $Q=\prod_{i=1}^{\infty}[-1,1]$ and the polar mapping is topologically conjugate with the standard based-free involution $σ:Q\rightarrow Q,$ defined by $σ(x)=-x$ for all $x\in Q.$ We also prove that among the inclusion-reversing involutions on $\mathcal K^n_0$ (also called dualities), those and only those with a unique fixed point are topologically conjugate with the polar mapping, and they can be characterized as all the maps $f:\mathcal{K}_0^n\to \mathcal{K}_0^n$ of the form $f(A)=T(A^{\circ})$, with $T$ a positive definite linear isomorphism of $\mathbb R^n$.

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arXiv2021

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.

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arXiv2021

Some notes on induced functions and group actions on hyperspaces

We discuss the problem of when a continuous map between topological spaces induces a continuous function between their respective hyperspaces. We characterize the continuity of the induced function in the case of the Fell and Attouch-Wets hyperspaces. Additionally we explore the problem of whether a continuous action of a topological group $G$ on a topological space $X$ induces a continuous action on $CL(X)$.

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arXiv2021

A Jordan Curve Theorem for 2-dimensional Tilings

The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plane is a quotient space of $\mathbb R^2$ induced by a tiling of squares, it is natural to ask for which other tilings of the plane it is possible to obtain a similar result. In this paper we prove a Jordan curve theorem which is valid for every locally finite tiling of $\mathbb R^2$. As a corollary of our result, we generalize some classical Jordan curve theorems for grids of points, including Rosenfeld's theorem.

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arXiv2019

Separation axioms and covering dimension of asymmetric normed spaces

In this paper, we approach the question if some of the separation axioms are equivalent in the class of asymmetric normed spaces. In particular, we make a remark on a known theorem which states that every $T_1$ asymmetric normed space with compact closed unit ball must be finite-dimensional. We also explore the product structure of these spaces and characterize the topological (covering) dimension of all finite-dimensional asymmetric normed spaces.

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arXiv2017

Local compactness in right bounded asymmetric normed spaces

We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric normed spaces. In the positive direction, we will prove that a finite dimensional asymmetric normed space is strongly locally compact if and only if it is right bounded.

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arXiv2016

A short proof of Grünbaum's Conjecture about affine invariant points

Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$, where Aff(n) denotes the group of all nonsingular affine maps of $\mathbb R^n$. For every $K\in\mathcal K_n$, let $\mathfrak{P}_n(K)=\{p(K)\in\mathbb R^n\mid p\text{ is an affine invariant point}\}$ and $\mathfrak{F}_n(K)=\{x\in\mathbb R^n\mid gx=x\text{ for every }g\in Aff(n)\text{ such that }gK=K\}$. In 1963, B. Grünbaum conjectured that $\mathfrak{P}_n(K)=\mathfrak{F}_n(K)$ . After some partial results, the conjecture was recently proven by O. Mordhorst. In this short note we give a rather different, simpler and shorter proof of this conjecture, based merely on the topology of the action of Aff(n) on $\mathcal K_n$.

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arXiv2015

A pseudometric invariant under similarities in the hyperspace of non-degenerated compact convex sets of $\mathbb R^n$

In this work we define a new pseudometric in $\mathcal K^n_*$, the hyperspace of all non-degenerated compact convex sets of $\mathbb R^n$, which is invariant under similarities. We will prove that the quotient space generated by this pseudometric (which is the orbit space generated by the natural action of the group of similarities on $\mathcal K^n_*$) is homeomorphic to the Banach-Mazur compactum $BM(n)$, while $\mathcal K^n_*$ is homeomorphic to the topological product $Q\times\mathbb R^{n+1}$, where $Q$ stands for the Hilbert cube. Finally we will show some consequences in convex geometry, namely, we measure how much two convex bodies differ (by means of our new pseudometric) in terms of some classical functionals.

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arXiv2014

Compact convex sets in 2-dimensional asymmetric normed lattices

We study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex sets with non empty interior in (R^2, q), where q is an asymmetric lattice norm.

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arXiv2014

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.

math.FA↗
arXiv2013

Hyperspaces of convex bodies of constant width

Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cw_D(R^n) of all compact convex subsets with constant width d\in D, where D is a convex subset of [0,\infty). Our main result states that cw_D(R^n) is homeomorphic to D\times R^n\times Q, where Q denotes the Hilbert cube. We also prove that the hyperspace crw_D(R^n), consisting of all pairs of compact convex sets of constant relative width d\in D is homeomorphic to cw_D(R^n). In particular, we prove that the hyperspace cw(R^n) of all compact convex bodies of constant width, as well as the hyperspace crw(R^n) of all pairs of compact convex sets of constant relative positive width are homeomorphic to R^{n+1}\times Q.

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