Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.28862

Kirszbraun's Intersection Property and Nonexpansive Extensions

Valentin Gutev
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper aims to present the historical development of the extension problem of nonexpansive maps in Euclidean and Hilbert spaces, and to serve as a comprehensive guide to this specific area. Such extensions are based on a geometric property of the intersection of closed balls in Euclidean spaces, known as the Kirszbraun intersection property. Subsequently, a number of authors have extended this property to Hilbert and other spaces. All these results are discussed in this paper, but the paper itself is not a mechanical compilation of already known proofs. The material has been developed with an emphasis on the natural connections between all these results and the corresponding concepts involved in them. Furthermore, most of these results are presented with new simplified proofs. Last but not least, the paper also contains some new results related to this extension problem.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Valentin Gutev. 2026-10-20. Kirszbraun's Intersection Property and Nonexpansive Extensions. https://arxiv.org/abs/2610.28862

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Sharp Planar Fractional Isoperimetric Inequality

In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $Ω\subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(Ω) \le \frac{π^{s-\frac{1}{2}}Γ\left(\frac{3-s}{2}\right)}{s(1-s)Γ\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partialΩ)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $Γ$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.

math.FA↗
arXiv2026

Norms of the Hilbert Matrix Operator on Some Analytic Function Spaces

For $\mathcal H:\mathcal B^α\to\mathcal B^α$ with $1<α<2$, we obtain an exact norm representation. In particular, the norm is given by an explicit Gamma-function formula for $1<α\leq3/2$, whereas for $3/2<α<2$ it strictly exceeds the corresponding Gamma-function expression. For $\mathcal H:\mathcal B^α\to\mathcal B^α_{\log}$, we determine the exact norm: it is equal to $3/2$ for $1<α\leq4/3$ and is given by an explicit Gamma-function formula for $4/3<α<2$. We also determine the exact norm of $\mathcal H:H^1\to H^1_{\log}$ and show that it is equal to $π$. Finally, for $0<α<1$, we determine the exact norm of $\mathcal H:H^\infty_α\to\mathcal B^{α+1}$, which is given by an explicit Gamma-function formula.

math.FA↗
arXiv2026

Second quantization and compactness of induced Gaussian quantum Markov semigroups

We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert-Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.

math.FA↗