On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls

摘要

We consider the outer approximation problem of finding a minimum radius ball enclosing a given intersection of at most n - 1 balls in R~n. We show that if the aforementioned intersection has a nonempty interior, then the problem reduces to minimizing a convex quadratic function over the unit simplex. This result is established by using convexity and representation theorems for a class of quadratic mappings. As a byproduct of our analysis, we show that a class of nonconvex quadratic problems admits a tight semidefinite relaxation.