 John ellipsoid

The John ellipsoid of a convex body is its inscribed ellipsoid of largest volume. For the unit ball $K$ of a $d$-dimensional normed vector space, change coordinates so that this ellipsoid is the Euclidean unit ball. The resulting coordinates are called John position. In these coordinates the John contact decomposition implies $B_2^d\subseteq K\subseteq \sqrt{d}{B}_2^d$, or $\Vert x{\Vert }_E\le |x|\le \sqrt{d}\Vert x{\Vert }_E$.

The John ellipsoid is a specific type of ellipsoid that is used in the context of convex analysis and optimization. It is associated with the John’s theorem, which deals with the geometry of convex bodies. More formally, the John ellipsoid of a convex body \( K \) in \( \mathbb{R}^n \) is the unique ellipsoid of maximal volume that can be inscribed in \( K \).