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Let S be a compact connected subset of Rd. A necessary and sufficient condition is given to ensure that the dimension of the convex kernel of S is greater than or equal to k, O<k<d. This condition involves a visibility constraint on the points of local nonconvexity of S. As consequences, we obtain new characterizations of the convex kernel of S and the nth-order convex kernel of S.

The dimension of the convex kernel and points of local nonconvexity