  The Gaussian width is a fundamental quantity in probability, statistics and
geometry, known to underlie the intrinsic difficulty of estimation and
hypothesis testing. In this work, we show how the Gaussian width, when
localized to any given point of an ellipse, can be controlled by the Kolmogorov
width of a set similarly localized. This connection leads to an explicit
characterization of the estimation error of least-squares regression as a
function of the true regression vector within the ellipse. The rate of error
decay varies substantially as a function of location: as a concrete example, in
Sobolev ellipses of smoothness $\alpha$, we exhibit rates that vary from
$(\sigma^2)^{\frac{2 \alpha}{2 \alpha + 1}}$, corresponding to the classical
global rate, to the faster rate $(\sigma^2)^{\frac{4 \alpha}{4 \alpha + 1}}$.
We also show how the local Kolmogorov width can be related to local metric
entropy.
