arXiv subjects

Giuliano Basso

Publications and source records attributed to Giuliano Basso.

17 recordsLinked to original sources

arXiv2024

Erratum to "Computation of maximal projection constants"

This is an erratum to the article: "Computation of maximal projection constants" (J. Funct. Anal., 277). The statement of Lemma 3.1(2) of that paper is incorrect. As a consequence of this the proof of Theorem 1.4 is incomplete. In this erratum we prove a corrected version of Lemma 3.1 and explain why, with this weaker result, our original strategy for proving Theorem 1.4 no longer works. The other results of the article, in particular the alternative proof of Gr\"unbaum's conjecture, do not depend on Lemma 3.1 and are thus not affected by this error.

math.FA↗

arXiv2023

Lipschitz extension theorems with explicit constants

In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows: If $X$ is any metric space and $A\subset X$ satisfies the condition $\text{Nagata}(n, c)$, then any $1$-Lipschitz map $f\colon A \to Y$ to a Banach space $Y$ admits a Lipschitz extension $F\colon X \to Y$ whose Lipschitz constant is at most $1000\cdot (c+1)\cdot \log_2(n+2)$. By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if $A\subset X$ consists of $n$ points, then Lipschitz extensions as above exist with a Lipschitz constant of at most $600 \cdot \log n \cdot (\log \log n)^{-1}$.

math.MG↗

arXiv2023

Geometric and analytic structures on metric spaces homeomorphic to a manifold

We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. Using our existence result, we establish that Riemannian manifolds are Lipschitz-volume rigid among certain metric manifolds and we show the validity of (relative) isoperimetric inequalities in metric $n$-manifolds that are Ahlfors $n$-regular and linearly locally contractible. The former statement is a generalization of a well-known Lipschitz-volume rigidity result in Riemannian geometry and the latter yields a relatively short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincaré inequalities in these spaces. Finally, as a further application, we also give sufficient conditions for a metric manifold to be rectifiable.

math.MG↗

arXiv2023

Conical geodesic bicombings on subsets of normed vector spaces

We prove existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a non-consistent convex geodesic bicombing. Furthermore, we show that under a mild geometric assumption on the norm, a conical bicombing on an open subset of a normed space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan-Hadamard type result that if a closed convex subset of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by linear segments.

math.MG↗

arXiv2023

On a problem of Mazur and Sternbach

We investigate Problem 155 form the "Scottish Book" due to S. Mazur and L. Sternbach. In modern terminology they asked if every bijective, locally isometric map between two real Banach spaces is always a global isometry. Recently, an affirmative answer when the source space is separable was obtained by M. Mori, using techniques related to the Mazur-Ulam theorem and its generalizations. The main purpose of this short note is to offer a different approach to Problem 155 motivated by recent advances in metric geometry. We show that it has an affirmative answer under the additional assumption that the map considered is a local isometry.

math.FA↗

arXiv2023

Finsler currents

We propose a slight variant of Ambrosio and Kirchheim's definition of a metric current. We show that with this new definition it is possible to obtain certain volume functionals from Finsler geometry as mass measures of currents. As an application, we obtain a whole family of extendibly convex $n$-volume densities. This family includes the mass* and the circumscribed Riemannian volume densities.

math.DG↗

arXiv2023

Integral representation of functions on the circle

We give a complete characterization of all real-valued functions on the unit circle $S^1$ that can be represented by integrating the spherical distance on $S^1$ with respect to a signed measure or a probability measure.

math.CA↗

arXiv2023

A non-compact convex hull in generalized non-positive curvature

In this article, we are interested in metric spaces that satisfy a weak non-positive curvature condition in the sense that they admit a conical geodesic bicombing. We show that the analog of a question of Gromov about compactness properties of convex hulls has a negative answer in this setting. Specifically, we prove that there exists a complete metric space $X$ that admits a conical bicombing $\sigma$ such that $X$ has a finite subset whose closed $\sigma$-convex hull is not compact.

math.MG↗

arXiv2022

Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

In this paper we consider metric fillings of convex bodies. We show that convex bodies $C\subset \mathbb{R}^n$ are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As a further application of this result, we answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.

math.DG↗

arXiv2022

Approximating spaces of Nagata dimension zero by weighted trees

We prove that if a metric space $X$ has Nagata dimension zero with constant $c$, then there exists a dense subset of $X$ that is $8c$-bilipschitz equivalent to a weighted tree. The factor $8$ is the best possible if $c=1$, that is, if $X$ is an ultrametric space. This yields a new proof of a result of Chan, Xia, Konjevod and Richa. Moreover, as an application, we also obtain quantitative versions of certain metric embedding and Lipschitz extension results of Lang and Schlichenmaier. Finally, we prove a variant of our main theorem for $0$-hyperbolic proper metric spaces. This generalizes a result of Gupta.

math.MG↗

arXiv2021

Undistorted fillings in subsets of metric spaces

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.

math.MG↗

arXiv2021

Absolute Lipschitz extendability and linear projection constants

We prove that the absolute extendability constant of a finite metric space may be determined by computing relative projection constants of certain Lipschitz-free spaces. As an application, we show that $\mbox{ae}(3)=4/3$ and $\mbox{ae}(4)\geq (5+4\sqrt{2})/7$. Moreover, we discuss how to compute relative projection constants by solving linear programming problems.

math.MG↗

arXiv2020

Extending and improving conical bicombings

We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric spaces admitting a conical bicombing by showing that every such space is isometric to a closed $\sigma$-convex subset of some injective metric space. In addition, we show that every proper metric space that admits a conical bicombing also admits a consistent bicombing that satisfies certain convexity conditions. This can be seen as a strong indication that a question from Descombes and Lang about improving conical bicombings might have a positive answer. As an application, we prove that any group acting geometrically on a proper metric space with a conical bicombing admits a $\mathcal{Z}$-structure.

math.MG↗

arXiv2020

Almost minimal orthogonal projections

The projection constant $Π(E):=Π(E, \ell_\infty)$ of a finite-dimensional Banach space $E\subset\ell_\infty$ is by definition the smallest norm of a linear projection of $\ell_\infty$ onto $E$. Fix $n\geq 1$ and denote by $Π_n$ the maximal value of $Π(\cdot)$ amongst $n$-dimensional real Banach spaces. We prove for every $\varepsilon >0$ that there exist an integer $d\geq 1$ and an $n$-dimensional subspace $E\subset\ell_1^d$ such that $Π_n \leq Π(E, \ell_1^d) +2 \varepsilon$ and the orthogonal projection $P\colon \ell_1^d\to E$ is almost minimal in the sense that $\lVert P \rVert \leq Π(E, \ell_1^d)+\varepsilon$. As a consequence of our main result, we obtain a formula relating $Π_n$ to smallest absolute value row-sums of orthogonal projection matrices of rank $n$.

math.FA↗

arXiv2020

Lipschitz extensions to finitely many points

We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

math.MG↗

arXiv2019

Computation of maximal projection constants

The linear projection constant $Π(E)$ of a finite-dimensional real Banach space $E$ is the smallest number $C\in [0,+\infty)$ such that $E$ is a $C$-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by $Π_n$ the maximal linear projection constant amongst $n$-dimensional Banach spaces. In this article, we prove that $Π_n$ may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the relative projection constants of codimension $n$ converge to $1+Π_n$. Furthermore, using the classification of $K_4$-free two-graphs, we give an alternative proof of $Π_2=\frac{4}{3}$. We also show by means of elementary functional analysis that for each integer $n\geq 1$ there exists a polyhedral $n$-dimensional Banach space $F_n$ such that $Π(F_n)=Π_n$.

math.MG↗

arXiv2015

Fixed point theorems for metric spaces with a conical geodesic bicombing

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a 1-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.

math.MG↗