This work concerns a
quantitative version of Krasnosel’skii’s theorem in R2, and the following result
is obtained: Let S be a nonempty compact subset of R2 having n points
of local nonconvexity. Then the kernel of S contains an interval of radius
𝜀 > 0 if and only if every f(n) =max{4,2n} points of S see via S a common
interval of radius 𝜀. The number f(n) in the theorem is best possible for every
n ≧ 1.