Ellipsoid-fitting threshold conjecture

Let $n,d\geq 1$, and let $x_1,\ldots,x_n$ be independent random vectors with

$$x_i\sim\mathcal{N}(0,\mathrm{I}_d/d).$$

For an ellipsoid $\mathcal{E}$, say that it is an ellipsoid fit to $(x_i)_{i=1}^n$ if every point $x_i$ lies on its boundary. Ellipsoid-fitting threshold conjecture. For every $\varepsilon>0$,

$$\limsup_{d\to\infty}\frac{n}{d^2}\leq\frac{1-\varepsilon}{4}\quad\Longrightarrow\quad\lim_{d\to\infty}\mathbb{P}\bigl[\exists\,\mathcal{E}\text{ an ellipsoid fit to }(x_i)_{i=1}^n\bigr]=1,$$

and

$$\liminf_{d\to\infty}\frac{n}{d^2}\geq\frac{1+\varepsilon}{4}\quad\Longrightarrow\quad\lim_{d\to\infty}\mathbb{P}\bigl[\exists\,\mathcal{E}\text{ an ellipsoid fit to }(x_i)_{i=1}^n\bigr]=0.$$

The conjecture predicts a sharp transition at $n/d^2\to 1/4$ for fitting high-dimensional Gaussian points by the boundary of a centered ellipsoid. Earlier work identified this threshold and related ellipsoid fitting to the success of a convex relaxation in low-rank matrix decomposition; the sharp transition remains open.