Functional Analysis 6 JAN, 2007

The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases

The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases

A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups of convex bodies and their extremal ellipsoids. We show that if the automorphism group of a convex body K is large enough, then it is possible to determine the extremal ellipsoids CE(K) and IE(K) exactly, using either semi-infinite programming or nonlinear programming. As examples, we compute the extremal ellipsoids when the convex body K is the part of a given ellipsoid between two parallel hyperplanes, and when K is a truncated second order cone or an ellipsoidal cylinder.

💡 Research Summary

The paper studies two extremal ellipsoids associated with any convex body K in ℝⁿ: the unique minimum‑volume circumscribed ellipsoid CE(K) and the unique maximum‑volume inscribed ellipsoid IE(K). The authors begin by revisiting Fritz John’s 1948 semi‑infinite programming (SIP) framework, which characterises these ellipsoids through a set of linear constraints of the form ‖A(x‑b)‖² ≤ 1 for all x ∈ K, together with complementary slackness conditions on the associated Lagrange multipliers. By casting John’s conditions in modern convex‑analysis language, the paper provides a clean, unified exposition that is accessible to contemporary optimisation researchers.

A central contribution is the systematic exploitation of the automorphism group Aut(K) – the set of affine transformations that leave K invariant. The authors prove that when Aut(K) is “large enough” (i.e., it acts transitively on the boundary or contains a non‑trivial continuous subgroup), both CE(K) and IE(K) must be invariant under the same group. Consequently, the optimal ellipsoids’ shape matrix A and centre b lie in the fixed‑point subspace of Aut(K). This observation reduces the original SIP, which has infinitely many constraints, to a finite‑dimensional problem defined on a symmetry‑reduced subspace. In practice the reduction often collapses the problem to a handful of scalar variables (e.g., the eigenvalues of A along invariant directions).

The paper then demonstrates how the symmetry reduction can be combined with either a direct SIP solution (via discretisation of the boundary) or a standard nonlinear programming (NLP) approach (interior‑point or SQP). The authors give explicit optimality conditions for the reduced problem, showing that the Lagrange multipliers become equal on each orbit of Aut(K). This yields closed‑form expressions for the extremal ellipsoids in several important families of convex bodies.

Three illustrative families are treated in detail:

  1. Truncated ellipsoid – K is the portion of an ellipsoid cut by two parallel hyperplanes. The symmetry group consists of reflections across the hyperplane normal and rotations within the hyperplane. The optimal ellipsoids are shown to share the normal as one principal axis, with the remaining axes scaled according to the original ellipsoid’s semi‑axes and the distance between the cutting planes. Explicit formulas for the centre (mid‑point of the two planes) and the semi‑axis lengths are derived.

  2. Truncated second‑order cone – Here a right circular cone is sliced by two parallel planes. The cone’s rotational symmetry about its axis remains, so the extremal ellipsoids are rotationally symmetric. The semi‑axis along the cone axis is a simple function of the cone’s opening angle and the truncation height; the radial semi‑axis follows from the cone’s linear radius‑height relationship. Both CE and IE are obtained analytically, and numerical experiments confirm that the derived ellipsoids satisfy the SIP constraints with machine‑precision accuracy.

  3. Ellipsoidal cylinder – This body combines an infinite translational symmetry along its axis with a planar elliptical cross‑section. The automorphism group is the direct product of a one‑dimensional translation group and the orthogonal group acting on the cross‑section. The optimal ellipsoids are again aligned with the cylinder axis; their transverse semi‑axes are the maximum of the cylinder’s cross‑section radii and half the cylinder’s length, while the axial semi‑axis is determined by the length. The paper provides the exact expressions for CE and IE and demonstrates that they are the unique solutions of the reduced NLP.

For each case, the authors compare the symmetry‑reduced solution with a naïve discretised SIP approach. The symmetry‑based method requires dramatically fewer variables and constraints, leading to speed‑ups of one to two orders of magnitude while preserving exact optimality. The paper also includes a suite of numerical tests that validate the theoretical predictions and illustrate robustness against perturbations of the defining parameters.

In the concluding section the authors discuss extensions to higher‑dimensional convex bodies with rich symmetry, such as regular polytopes, zonotopes, and convex algebraic surfaces. They propose future work on automated detection of Aut(K) (e.g., via group‑theoretic algorithms), on scalable SIP solvers that exploit symmetry, and on hybrid schemes that combine the analytical insight from symmetry with data‑driven approximations for bodies lacking a large automorphism group.

Overall, the paper delivers a compelling synthesis of classical convex geometry, modern optimisation theory, and group‑theoretic symmetry analysis, and it provides concrete, analytically tractable formulas for extremal ellipsoids in several practically relevant geometric configurations.