# A generalization of L"owner-John's ellipsoid theorem

arXiv: https://arxiv.org/abs/1302.1056 · HTML: https://mlsanity.com/paper/1302.1056/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2013.

## Built on by (most cited first, 1 of 1)

- [Uncertainty Quantification for Visual Object Pose Estimation: S-Lemma Ellipsoidal Bounds](https://mlsanity.com/paper/2511.21666/) (2025)

## Abstract (arXiv)

We address the following generalization P of the Lowner-John ellipsoid problem. Given a (non necessarily convex) compact set Ksubset Rn and an even integer d, find an homogeneous polynomial g of degree d such that Ksubset G:=x:g(x)≤1 and G has minimum volume among all such sets. We show that P is a convex optimization problem even if neither K nor G are convex! We next show that P has a unique optimal solution and a characterization with at most n+d-1choose d contacts points in Kcap G is also provided. This is the analogue for dtextgreater2 of the Lowner-John's theorem in the quadratic case d=2, but importantly, we neither require the set K nor the sublevel set G to be convex. More generally, there is also an homogeneous polynomial g of even degree d and a point a∈ Rn such that Ksubset Ga:=x:g(x-a)≤1 and Ga has minimum volume among all such sets (but uniqueness is not guaranteed). Finally, we also outline a numerical scheme to approximate as closely as desired the optimal value and an optimal solution. It consists of solving a hierarchy of convex optimization problems with strictly convex objective function and Linear Matrix Inequality (LMI) constraints.