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In a normed linear space X, consider a nonempty closed set K which has the property that for some r > 0 there exists a set of points xo € X\K, d(xoK) > r, which have closest points p(xo) € K and where the set of points xo -r((xo -p(xo))/\\xo -p(zo)||) is dense in X\K. If the norm has sufficiently strong differentiability properties, then the distance function d generated by K has similar differentiability properties and it follows that, in some spaces, K is convex. Given a real normed lineardoi:10.2307/2046995 fatcat:oimglq3hxjblhi6bo33hsqi3mq