To develop some intuition, the following argument might help, suggested 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.

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.