  The condition number of a differentiable convex function, namely the ratio of
its smoothness to strong convexity constants, is closely tied to fundamental
properties of the function. In particular, the condition number of a quadratic
convex function is the square of the aspect ratio of a canonical ellipsoid
associated to the function. Furthermore, the condition number of a function
bounds the linear rate of convergence of the gradient descent algorithm for
unconstrained convex minimization.
  We propose a condition number of a differentiable convex function relative to
a reference convex set and distance function pair. This relative condition
number is defined as the ratio of a relative smoothness to a relative strong
convexity constants. We show that the relative condition number extends the
main properties of the traditional condition number both in terms of its
geometric insight and in terms of its role in characterizing the linear
convergence of first-order methods for constrained convex minimization.
  When the reference set $X$ is a convex cone or a polyhedron and the function
$f$ is of the form $f = g\circ A$, we provide characterizations of and bounds
on the condition number of $f$ relative to $X$ in terms of the usual condition
number of $g$ and a suitable condition number of the pair $(A,X)$.
