A decomposition theorem for χ1-convex sets

✍ Scribed by Marilyn Breen

• Publisher: Elsevier Science
• Year: 1982
• Tongue: English
• Weight: 830 KB
• Volume: 41
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(82)90199-6

✦ Synopsis

A set S in 1 :" is said to he X,-convex if and only if S does not contain a visually independent subset having cardinality h', . It is natural tts ask when an h',-convex set may be expressed as a countable unI.,n of convex sets. Here i! is proved that if S is a closed h',-convex set in the plane and W' -S has at most finitely mall!' 'noundeci components. then S is a countaWe union of convex sets. A parallel result holds in Wd when S is a closed K,-convex set which contains ail triangular regions whose relative boundaries al-e in S. However, the result fails for arbitrary K,-convex sets, even in the plane.