Clear visibility and sets which are almost starshaped

Breen, Marilyn

doi:10.1090/S0002-9939-1984-0746099-5

Clear visibility and sets which are almost starshaped
HTML articles powered by AMS MathViewer

by Marilyn Breen PDF

Proc. Amer. Math. Soc. 91 (1984), 607-610 Request permission

Abstract:

Let $S$ be a nonempty compact connected set in ${R^2}$. Assume that every two points of local nonconvexity of $S$ are clearly visible from a common point of $S$. Then $S$ is "almost" starshaped. In fact, for any point $y$ in ${R^2}$ there is a line $L$ through $y$ such that $L \cap S$ is convex and every point of $S$ sees via $S$ a point of $L$. If "clearly visible" is replaced by the weaker term "visible", then the result fails.

References

Marilyn Breen, A Krasnosel′skiĭ-type theorem for points of local nonconvexity, Proc. Amer. Math. Soc. 85 (1982), no. 2, 261–266. MR 652454, DOI 10.1090/S0002-9939-1982-0652454-2
Marilyn Breen, ${\cal L}_{2}$ sets which are almost starshaped, Geometriae Dedicata 6 (1977), no. 4, 485–494. MR 470866, DOI 10.1007/BF00147785
Alfred Horn, Some generalization of Helly’s theorem on convex sets, Bull. Amer. Math. Soc. 55 (1949), 923–929. MR 31759, DOI 10.1090/S0002-9904-1949-09309-6
M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248
Steven R. Lay, Convex sets and their applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 655598
Heinrich Tietze, Über Konvexheit im kleinen und im großen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928), no. 1, 697–707 (German). MR 1544985, DOI 10.1007/BF01181191
Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
F. A. Valentine, Local convexity and $L_{n}$ sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310. MR 185510, DOI 10.1090/S0002-9939-1965-0185510-6