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On Sets that Meet Every Hyperplane in n ‐Space in at Most n Points
Author(s) -
Dijkstra Jan J.,
Van Mill Jan
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008979
Subject(s) - mathematics , hyperplane , plane (geometry) , simple (philosophy) , line (geometry) , point (geometry) , set (abstract data type) , space (punctuation) , nowhere dense set , combinatorics , euclidean space , euclidean geometry , assertion , geometry , computer science , philosophy , epistemology , programming language , operating system
A simple proof that no subset of the plane that meets every line in precisely two points is an F σ ‐set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero‐dimensional, and that if it is σ‐compact then it must be a nowhere dense G δ ‐set in the plane. Generalizations for similar sets in higher‐dimensional Euclidean spaces are also presented.