To develop some intuition, the following argument might help, suggested (and dismissed) by K. Villaverde, O. Kosheleva, and M. Ceberio, Why Ellipsoid Constraints, Ellipsoid Clusters, and Riemannian Space-Time: Dvoretzky's Theorem Revisited.
A stronger version of Dvoretzky’s theorem (due to Milman) asserts that almost all low-dimensional sections of a convex set have an almost ellipsoidal shape. An $n$-dimensional section consists of points $(x_1,x_2,\ldots x_n)$ such that $g(x_1,x_2,\ldots x_n)\leq 0$. Generically, this function $g$ will be smooth and a Taylor expansion to second order would be a good approximation, $$\sum_{i,j=1}^n a_{ij}x_i x_j+\sum_{i=1}^n b_i x_i \leq a_0,$$ producing an ellipsoid.