English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975)proved that then, for almost all $r>0$, the level set (distance sphere,$r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological$(n-1)$-dimensional manifold. This result was improved by J.H.G. Fu (1985). Weshow that Ferry's result is an easy consequence of the only fact that thedistance function $d(x)= \dist(x,F)$ is locally DC and has no stationary pointin $\R^n\setminus F$. Using this observation, we show that Ferry's (and evenFu's) result extends to sufficiently smooth normed linear spaces $X$ with $\dimX \in \{2,3\}$ (e.g., to $\ell^p_n, n=2,3, p\geq 2$), which improves andgeneralizes a result of R. Gariepy and W.D. Pepe (1972). By the same method wealso generalize Fu's result to Riemannian manifolds and improve a result of K.Shiohama and M. Tanaka (1996) on distance spheres in Alexandrov spaces.

Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces