Fitting Gaussian vectors in normed spaces

Given a family of $N$ i.i.d. Gaussian points in a $d$-dimensional normed space, what is the probability that some linear transformation maps it to the unit sphere? The case of a Euclidean space is known as the ellipsoid fitting problem and has recently attracted major attention. We initiate a study of this fitting problem in a general normed space, both asymptotically and in a fixed dimension. We specifically focus on the space ${\mathrm{\ell }}_{\mathrm{\infty }}^d$, where a combinatorial reformulation allows us to show that $c{d}^2\mathrm{/}\log d$ points can be fitted with high probability. We illustrate that the geometry of the normed space is reflected in the fitting problem by showing that in dimension $2$, the spaces ${\mathrm{\ell }}_2^2$ and ${\mathrm{\ell }}_{\mathrm{\infty }}^2$ are characterized by having extremal fittability properties.