Differentiability of Distance Functions and a Proximinal Property Inducing Convexity

J. R. Giles
1988 Proceedings of the American Mathematical Society  
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 linear
more » ... al normed linear space X, a subset K is called a proximinal ( Chebyshev) set if for each x E X\K there exists a (unique) p(x) E K such that
doi:10.2307/2046995 fatcat:oimglq3hxjblhi6bo33hsqi3mq