Conic version of Loewner–John ellipsoid theorem

We extend John’s inscribed ellipsoid theorem, as well as Loewner’s circumscribed ellipsoid theorem, from convex bodies to proper cones. To be more precise, we prove that a proper cone $$K$$K in $$\mathbb {R}^n$$Rn contains a unique ellipsoidal cone $$Q^\mathrm{in}(K)$$Qin(K) of maximal canonical volume and, on the other hand, it is enclosed by a unique ellipsoidal cone $$Q^\mathrm{circ}(K)$$Qcirc(K) of minimal canonical volume. In addition, we explain how to construct the inscribed ellipsoidal cone $$Q^\mathrm{in}(K)$$Qin(K). The circumscribed ellipsoidal cone $$Q^\mathrm{circ}(K)$$Qcirc(K) is then obtained by duality arguments. The canonical volume of an ellipsoidal cone is defined as the usual $$n$$n-dimensional volume of a certain truncation of the cone.