arXiv subjects

Franz Luggin

Publications and source records attributed to Franz Luggin.

4 recordsLinked to original sources

arXiv2025

Isometric Representation of Lipschitz-Free Spaces over Connected Orientable Riemannian Manifolds

We show that the Lipschitz-Free Space over a connected orientable $n$-di\-men\-sio\-nal Riemannian manifold $M$ is isometrically isomorphic to a quotient of $L^1(M,TM)$, the integrable sections of the tangent bundle $TM$, if $M$ is either complete or lies isometrically inside a complete manifold $N$. Two functions are deemed equivalent in this quotient space if their difference has distributional divergence zero. This quotient is the pre-annihilator of the exact essentially bounded currents, and if $M$ is simply connected, one may replace ``exact'' with ``closed'' currents.

math.FA↗

arXiv2024

Tilings of the Hyperbolic Space and Lipschitz Functions

We use a special tiling for the hyperbolic $d$-space $\mathbb{H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(P)\oplus\mathcal{F}(\mathcal{N})$ where $P$ is a polytope in $\mathbb{R}^d$ and $\mathcal{N}$ a net in $\mathbb{H}^d$ coming from the tiling. This implies that the spaces $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(\mathbb{R}^{d})\oplus \mathcal{F}(\mathcal{M})$ are isomorphic for every net $\mathcal{M}$ in $\mathbb{H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal{F}(\mathbb{H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm{Lip}(\mathbb{H}^{d})$ and $\mathrm{Lip}(\mathbb{R}^d)$.

math.FA↗

arXiv2023

Observations on the metric projection in finite dimensional Banach spaces

We consider the method of alternating (metric) projections for pairs of linear subspaces of finite dimensional Banach spaces. We investigate the size of the set of points for which this method converges to the metric projection onto the intersection of these subspaces. In addition we give a characterisation of the pairs of subspaces for which the alternating projection method converges to the projection onto the intersection for every initial point. We provide a characterisation of the linear subspaces of $\ell_p^n$, $1<p<\infty$, $p\neq 2$, which admit a linear metric projection and use this characterisation to show that in $\ell_p^3$, $1<p<\infty$, $p\neq 2$, the set of pairs of subspaces for which the alternating projection method converges to the projection onto the intersection is small in a probabilistic sense.

math.FA↗

arXiv2023

Homogeneous isosceles-free spaces

We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to isometry as vector spaces over the two-element field, endowed with an injective norm. Using isosceles-free decompositions, we provide bounds on the maximal number of distances in arbitrary homogeneous finite metric spaces.

math.LO↗