The $1/n$ ball lies in the hull

The geometric core of Löwner–John rounding: a John decomposition of the identity forces the ball of radius $1/n$ to lie inside the convex hull of the points.

Let $n \ge 1$, let $x_1,\dots,x_m \in \mathbb{R}^n$, and let $\lambda_1,\dots,\lambda_m \ge 0$ satisfy the normalized optimality identities

where $x_i x_i^{T}$ is the rank-one outer product and $I$ the $n \times n$ identity; the third family is complementary slackness, so every $x_i$ carrying a nonzero weight lies on the unit sphere. Then every $v \in \mathbb{R}^n$ whose Euclidean norm satisfies $\lVert v \rVert_2 \le 1/n$, i.e. $v^{T} v \le 1/n^{2}$, lies in the hull of the points:

This is the step that produces the dimension-dependent constant in Löwner–John rounding, and it is stated separately from the optimization problem so that it applies to any family satisfying a John decomposition of the identity, whatever its origin — the same statement is the workhorse behind John's theorem in Banach-space geometry. The constant is sharp: for the regular simplex inscribed in the unit sphere, the largest centred ball inside the simplex has radius exactly $1/n$.

Formalization Note Both norms are written through the dot product ⬝ᵥ on Fin n → ℝ, so the hypothesis on $v$ reads v ⬝ᵥ v ≤ 1 / n ^ 2; the dimension is called nn and is assumed positive. The fourth KKT identity $\sum_i \lambda_i = n$ is not needed and is deliberately absent from the hypotheses. Source: Boyd & Vandenberghe §8.4.1, pp. 412–413.