  The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo and
Willsky considers the maximum number $n$ random Gaussian points in
$\mathbb{R}^d$, such that with high probability, there exists an
origin-symmetric ellipsoid passing through all the points. They conjectured a
threshold of $n = (1-o_d(1)) \cdot d^2/4$, while until recently, known lower
bounds on the maximum possible $n$ were of the form $d^2/(\log d)^{O(1)}$. We
give a simple proof based on concentration of sample covariance matrices, that
with probability $1 - o_d(1)$, it is possible to fit an ellipsoid through
$d^2/C$ random Gaussian points. Similar results were also obtained in two
recent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023]
and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].
