Matches in SemOpenAlex for { <https://semopenalex.org/work/W3128061637> ?p ?o ?g. }
Showing items 1 to 67 of
67
with 100 items per page.
- W3128061637 abstract "For a family $mathcal{F}$ of sets in $mathbb{R}^d$, the Krasnoselskii number of $mathcal{F}$ is the smallest $m$ such that for any $S in mathcal{F}$, if every $m$ points of $S$ are visible from a common point in $S$, then any finite subset of $S$ is visible from a single point. More than 35 years ago, Peterson asked whether there exists a Krasnoselskii number for general sets in $mathbb{R}^d$. Excluding results for special cases of sets with strong topological restrictions, the best known result is due to Breen, who showed that if such a Krasnoselskii number in $mathbb{R}^2$ exists, then it is larger than $8$. In this paper we answer Peterson's question in the negative by showing that there is no Krasnoselskii number for the family of all sets in $mathbb{R}^2$. The proof is non-constructive, and uses transfinite induction and the well ordering theorem. In addition, we consider Krasnoselskii numbers with respect to visibility through polygonal paths of length $ leq n$, for which an analogue of Krasnoselskii's theorem was proved by Magazanik and Perles. We show, by an explicit construction, that for any $n geq 2$, there is no Krasnoselskii number for the family of general sets in $mathbb{R}^2$ with respect to visibility through paths of length $leq n$. (Here the counterexamples are finite unions of line segments.)" @default.
- W3128061637 created "2021-02-15" @default.
- W3128061637 creator A5005040583 @default.
- W3128061637 creator A5038150960 @default.
- W3128061637 date "2020-01-01" @default.
- W3128061637 modified "2023-09-27" @default.
- W3128061637 title "No Krasnoselskii number for general sets in R 2 ." @default.
- W3128061637 hasPublicationYear "2020" @default.
- W3128061637 type Work @default.
- W3128061637 sameAs 3128061637 @default.
- W3128061637 citedByCount "0" @default.
- W3128061637 crossrefType "journal-article" @default.
- W3128061637 hasAuthorship W3128061637A5005040583 @default.
- W3128061637 hasAuthorship W3128061637A5038150960 @default.
- W3128061637 hasConcept C111919701 @default.
- W3128061637 hasConcept C114614502 @default.
- W3128061637 hasConcept C118615104 @default.
- W3128061637 hasConcept C150591576 @default.
- W3128061637 hasConcept C162838799 @default.
- W3128061637 hasConcept C177264268 @default.
- W3128061637 hasConcept C199360897 @default.
- W3128061637 hasConcept C2524010 @default.
- W3128061637 hasConcept C2778701210 @default.
- W3128061637 hasConcept C28719098 @default.
- W3128061637 hasConcept C33923547 @default.
- W3128061637 hasConcept C41008148 @default.
- W3128061637 hasConcept C98045186 @default.
- W3128061637 hasConceptScore W3128061637C111919701 @default.
- W3128061637 hasConceptScore W3128061637C114614502 @default.
- W3128061637 hasConceptScore W3128061637C118615104 @default.
- W3128061637 hasConceptScore W3128061637C150591576 @default.
- W3128061637 hasConceptScore W3128061637C162838799 @default.
- W3128061637 hasConceptScore W3128061637C177264268 @default.
- W3128061637 hasConceptScore W3128061637C199360897 @default.
- W3128061637 hasConceptScore W3128061637C2524010 @default.
- W3128061637 hasConceptScore W3128061637C2778701210 @default.
- W3128061637 hasConceptScore W3128061637C28719098 @default.
- W3128061637 hasConceptScore W3128061637C33923547 @default.
- W3128061637 hasConceptScore W3128061637C41008148 @default.
- W3128061637 hasConceptScore W3128061637C98045186 @default.
- W3128061637 hasLocation W31280616371 @default.
- W3128061637 hasOpenAccess W3128061637 @default.
- W3128061637 hasPrimaryLocation W31280616371 @default.
- W3128061637 hasRelatedWork W2009961714 @default.
- W3128061637 hasRelatedWork W2036547219 @default.
- W3128061637 hasRelatedWork W2063885590 @default.
- W3128061637 hasRelatedWork W2241896673 @default.
- W3128061637 hasRelatedWork W2319187540 @default.
- W3128061637 hasRelatedWork W2754253106 @default.
- W3128061637 hasRelatedWork W2799144954 @default.
- W3128061637 hasRelatedWork W2799665955 @default.
- W3128061637 hasRelatedWork W2949325206 @default.
- W3128061637 hasRelatedWork W2962851451 @default.
- W3128061637 hasRelatedWork W2963297792 @default.
- W3128061637 hasRelatedWork W2964026705 @default.
- W3128061637 hasRelatedWork W2975981984 @default.
- W3128061637 hasRelatedWork W2990219750 @default.
- W3128061637 hasRelatedWork W3016883859 @default.
- W3128061637 hasRelatedWork W3105992356 @default.
- W3128061637 hasRelatedWork W3110728187 @default.
- W3128061637 hasRelatedWork W3121325671 @default.
- W3128061637 hasRelatedWork W3196314919 @default.
- W3128061637 hasRelatedWork W3209059903 @default.
- W3128061637 isParatext "false" @default.
- W3128061637 isRetracted "false" @default.
- W3128061637 magId "3128061637" @default.
- W3128061637 workType "article" @default.