Uniform Strong Proximinality and Continuity of Metric Projection

We present a sufficient condition for the uniform continuity of metric projection. This condition is a natural strengthening of the notion of strong proximinality, appearing in the literature of the past few years. We show that this condition is equivalent to the sufficient condition of continuity of metric projection introduced by K.-S. Lau back in 1979. A characterization of uniform convexity through proximinality is presented and we also relate quantitatively the power type estimate of modulus of uniform strong proximinality to the power type estimate of modulus of uniform convexity.