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Let S be a compact simply connected set in the plane. Let K(n) denote the generalized convex kernel of S of order n, bd S the boundary of S, E (0, S) the set of 0-extreme points of S, and for x E S, let K(n, x) denote the nth order convex kernel of x in S. It is known that K(n) = n {K(n, x)|x E bd S) and in certain cases, it is known that K(1) = n {K(1, x)|x E E(O, S)). The main result of this paper extends the above mentioned results for certain sets. It is shown that K(n) = n {K(n, x)lx E E(O, S)) for certain compact simply connected sets S in the plane. In the process of obtaining this result, a characterization of the boundary is also obtained.

Zero-extreme points and the generalized convex kernel