A Weak Krasnosel'skii Theorem in R d

Abstract

Let S be a compact, locally starshaped set in Rd, and let k be a fixed integer, 0 0, x is 6-visible from y via S if and only if y sees via S each point of N(x, 6) n s, where N(x, 6) is the neighborhood of x having radius 6. Set S is starshaped if and only if there is some point p of S such that p sees each point of S via S, and the set of all such points p is called the (convex) kernel of S, denoted ker S. Finally, set S is locally starshaped at point s if and only if for some neighborhood N of s, N n s is starshaped with p E ker(Nn s). A well-known theorem of Krasnosel'skii [9] states that if S is a nonempty compact set in Rd, S is starshaped if and only if every d + 1 points of S are visible via S from a common point. The proof is based on the following theorem of Helly [4]: If 9 is a family of compact sets in Rd, n{F: F in F} :A 0 if and only if every d + 1 members of F have a nonempty intersection. Various analogues of Helly's theorem have been found when the number d+ 1 is replaced by a smaller number. (See Horn and Valentine [6], Horn [5], Klee [7].) In particular, Horn [5] has proved that for 1 < k < di if every d k + 1 members of F have a nonempty intersection, then for any (k 1)-flat F1, there corresponds a k-flat F2 containing F1 and meeting every member of S. Klee [8] has pointed out that it would be interesting to investigate corresponding weak analogues of Krasnosel'skii's theorem as well. In [1], Horn's theorem and the notion of clearly visible were used to obtain the following result: Let S be a nonempty compact connected set in R2, and assume that every two points of local nonconvexity of S are clearly visible via S from a common point. Then for any point y in R2, there is a line L through y such that L n s is convex and every point of S sees via S a point of L. Here a similar result is obtained in higher dimensions: Let S be a compact, locally starshaped set in Rd, Received by the editors March 5, 1987 and, in revised form, November 2, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 52A35, 52A30. (E)1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page