  We consider the problem of chasing convex functions, where functions arrive
over time. The player takes actions after seeing the function, and the goal is
to achieve a small function cost for these actions, as well as a small cost for
moving between actions. While the general problem requires a polynomial
dependence on the dimension, we show how to get dimension-independent bounds
for well-behaved functions. In particular, we consider the case where the
convex functions are $\kappa$-well-conditioned, and give an algorithm that
achieves an $O(\sqrt \kappa)$-competitiveness. Moreover, when the functions are
supported on $k$-dimensional affine subspaces--e.g., when the function are the
indicators of some affine subspaces--we get $O(\min(k, \sqrt{k \log
T}))$-competitive algorithms for request sequences of length $T$. We also show
some lower bounds, that well-conditioned functions require
$\Omega(\kappa^{1/3})$-competitiveness, and $k$-dimensional functions require
$\Omega(\sqrt{k})$-competitiveness.
