Existence of the Löwner–John ellipsoid

Theorem. For all natural numbers $nn$ and $m$ and every family of $m$ points $x_1,\dots,x_m\in\mathbb{R}^{nn}$ (a function $x:\mathrm{Fin}\,m\to(\mathrm{Fin}\,nn\to\mathbb{R})$; the set considered is its range, i.e. the set of the points with duplicates merged), if the topological interior of the convex hull over $\mathbb{R}$ of $\{x_i\}$ is nonempty (interior in the product topology on $\mathrm{Fin}\,nn\to\mathbb{R}$, which coincides with the usual Euclidean topology in finite dimension), then there exist a matrix $A$ and a vector $b$ such that: $A$ is symmetric; $A$ is positive definite (Mathlib's notion, which includes symmetry); every $x_i$ satisfies $\lVert Ax_i+b\rVert^{2}\le 1$; and for every symmetric positive-definite $A'$ and every offset $b'$ whose body $\{v:\lVert A'v+b'\rVert^{2}\le 1\}$ contains all the $x_i$, one has $\det A'\le\det A$. This is plain existence ($\exists$), not unique existence. Edge cases: if $m=0$ the range is empty, its convex hull is empty, its interior is empty, so the hypothesis fails and the statement is vacuous there. If $nn=0$ and $m\ge 1$ the ambient space is a single point, which is open, so the hypothesis holds and the conclusion concerns the unique $0\times 0$ matrix (determinant $1$). The nonempty-interior hypothesis is the full-dimensionality condition on the finite point family; nothing else constrains $m$ relative to $nn$.