The object of this paper is to first extend the definition of the uniform cusp property introduced in Reference 1 to a larger class of dominating cusp functions continuous only at the origin along with the W~(1,p)-compactness theorem for the family of all subsets of a bounded holdall verifying that property. The local C°-graphs of sets with a compact boundary verifying a segment property are further characterized, and such sets are shown to satisfy a uniform cusp property for a dominating non-negative cusp function that is continuous only at the origin. Those characterizations are used in the last section to present a new sufficient condition for the compactness of the family of subsets of a bounded holdall, which are locally C°-epigraphs and whose local C°-graphs are dominated by a single cusp function. Finally, a streamlined version of the sufficient condition of Reference 3 is also given as a special case of this condition.
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