Conic version of Loewner-John ellipsoid theorem (Q5962723)

For an elliptic cone \(K\) in \(\mathbb R^{n}\) with apex at the origin \(o\) and symmetry axis \(\text{pos}\{ s\}\) where \(s\in S^{n-1}\), the \textit{canonical volume} Vol~\(|K|\) is the volume of the set \(\{ x\in K:s\cdot x\leq 1\}\). The authors show that for each proper convex cone \(K\) in \(\mathbb R^{n}\) there are unique elliptic cones \(E\), \(F\) such that \(E\subseteq K\subseteq F\) where \(E\) has maximum and \(F\) minimum canonical volume. Further properties of \(E\), \(F\), including John-type characterizations are shown, too. For the existence and uniqueness of the ellipsoids \(E\), \(F\) see also [\textit{L. Danzer} et al., Arch. Math. 8, 214--219 (1957; Zbl 0078.35803)] and [\textit{V. L. Zaguskin}, Usp. Mat. Nauk 13, No. 6(84), 89--93 (1958; Zbl 0089.17403)]. A different approach to John type theorems is due to the referee and \textit{F. E. Schuster} [Arch. Math. 85, No. 1, 82--88 (2005; Zbl 1086.52002)].