John Ellipsoid and Its Functional Extensions

• John Ellipsoid is the unique maximal volume ellipsoid inscribed in a convex body, characterized by contact points and weighted decompositions.
• The functional John ellipsoid extends this classical concept to proper log-concave functions via s-liftings and variational principles.
• Constructive convex minimization methods yield stable, quantitative decompositions that facilitate numerical computation in high-dimensional convex analysis.

John Ellipsoid refers to the unique maximal volume ellipsoid inscribed in a convex body, as characterized by Fritz John’s classical theorem. Its rigorous definition, decomposition theory, algorithmic computation, and generalizations to functional and high-dimensional settings form a foundational pillar of modern convex geometry, optimization, and analysis. The John ellipsoid also admits powerful extensions in terms of contact decompositions and variational principles, as exemplified by the functional John ellipsoid introduced by Ivanov and Naszódi and further developed through constructive minimization methods.

1. Definition and Classical Decomposition

Given a convex body $K \subset \mathbb{R}^n$ with nonempty interior, the John ellipsoid $J(K)$ is the unique ellipsoid of largest volume contained in $K$. If $K$ is centrally symmetric, one can assume without loss of generality that the Euclidean unit ball $B_n = \{x \in \mathbb{R}^n : \|x\| \leq 1\}$ is the John ellipsoid, up to affine transformation. The fundamental characterization is as follows:

• There exist contact points $u_1, \dots, u_m$ in $\partial K \cap S^{n-1}$ and positive weights $c_1, \dots, c_m$ such that

$$\sum_{i=1}^m c_i u_i = 0, \qquad \sum_{i=1}^m c_i (u_i \otimes u_i) = I_n,$$

where $u_i \otimes u_i$ denotes the rank-one operator $x \mapsto (u_i \cdot x)u_i$.

This matrix decomposition, commonly called the "John decomposition of the identity," is essential for understanding isotropy, criticality, and the geometric representation of convex bodies in terms of their extremal ellipsoids (Gruber et al., 2012).

2. The Functional John Ellipsoid: Ivanov–Naszódi Theory

Ivanov and Naszódi introduced a fundamentally new generalization to the setting of proper log-concave functions $h : \mathbb{R}^n \rightarrow [0,\infty)$, where $h(x) = e^{-V(x)}$ for convex $V$ and $0 < \int h < \infty$. For each parameter $s > 0$, one defines the $s$-lifting: $$(s)h = \{ (x, t) \in \mathbb{R}^n \times \mathbb{R} : |t| \leq h(x)^{1/s} \},$$ and considers $n$-symmetric ellipsoids $E(A, a) \subset \mathbb{R}^{n+1}$ parametrized by positive-definite $A$ and $a > 0$. The $s$-volume of such an ellipsoid is

$$(s)\!\operatorname{vol}(E(A, a)) = \int_{E(A, a)} |t|^{s-1} d(x,t).$$

The John $s$-ellipsoid of $h$ is the unique $E(h, s)$ of maximal $s$-volume contained in $(s)h$. Its $s$-marginal (the John $s$-function) is

$$h_{J,s}(x) = \int_{-h(x)^{1/s}}^{h(x)^{1/s}} \mathbf{1}_{E(A_0,a_0)}(x,t) |t|^{s-1}dt,$$

which is again log-concave and satisfies $h_{J,s} \leq h^{1/s}$ (Ivanov et al., 2020, Baêta, 2 Apr 2025).

The key structural result is a functional John decomposition mirroring the classical setting:

For proper log-concave $h$ and $s > 0$, $h_{J,s}$ maximizes the lifted $s$-problem if and only if there exist finitely many "contact points" $u_i \in B_n$ and positive weights $c_i$ such that - $h(u_i) = h_{J,s}(u_i)$ for each $i$ - $\sum_i c_i (u_i \otimes u_i) = I_n$ - $\sum_i c_i u_i = 0$ - $\sum_i c_i h(u_i)^{1/s} = s$

This decomposition of the identity is the canonical convex-analytic certificate for extremality in the functional context (Baêta, 2 Apr 2025).

3. Constructive Decomposition via Convex Minimization

A major technical advance is the constructive realization of the decomposition for the functional John ellipsoid via explicit convex minimization. As established by Baêta,

• One introduces auxiliary one-variable convex functions $f, g$ with prescribed support properties.
• For each $r \in (\frac12, 1)$, one defines a strictly convex functional $L_r(A,a,w)$, whose minimization over the space of $n$-symmetric ellipsoid parameters yields a unique optimizer.
• The corresponding minimizer determines a centered, isotropic, finitely supported measure supported on the set where $h(x) = h_{J,s}(x)$.
• In the limit $r \to 1^-$, this measure converges (in the sense of measures) to an atomic measure supported on the contact points, realizing the required decomposition.

The method is robust, admits quantitative stability, and bounds the number of contact points by Carathéodory’s theorem: at most $(n+1)(n+2)/2$ are needed (Baêta, 2 Apr 2025).

4. Comparison with Classical John Theory and Interpolation Phenomena

The functional setting recovers the classical John theory in the limit $s \to 0$, since

$h_{J,s} \xrightarrow{s \to 0} \beta_0 \chi_E,$

where $E$ is the maximal volume ellipsoid in the level set $\{ h \geq \beta_0 \}$. As $s \to \infty$, the maximizers are Gaussian densities of maximal integral dominated by $h$. Thus, the one-parameter family of functional John ellipsoids interpolates between the indicator-based geometric theory and the Gaussian-dominated analytic regime (Ivanov et al., 2020). The decomposition theorem, accordingly, passes to the sum

$$\sum_{i=1}^k c_i u_i \otimes u_i = I_n, \quad \sum_{i=1}^k c_i u_i = 0,$$

in the classical case, and generalizes to involve normalization factors in the functional case.

5. Applications, Stability, and Theoretical Significance

Uniqueness of the functional John ellipsoid follows from strict convexity of the volume functional (for fixed $s$), and the decomposition is stable under perturbations of the data. Applications include:

• Numerical schemes for explicit computation of functional John ellipsoids and their contact/support measures.
• Analytical bounds in reverse isoperimetric inequalities for log-concave measures.
• A powerful total-variational principle bridging convex geometric analysis and log-concave optimization.

Comparison with the classical John ellipsoid underlines the deeper insight offered by the functional setting, particularly in the ability to interpolate smoothly between geometric and analytic extremal objects.

6. Broader and Ongoing Context

The functional John ellipsoid framework has catalyzed further research in:

• The geometry of log-concave functions (Chen et al., 2021, Ivanov, 24 Dec 2024), offering deeper understanding of volume ratios, inclusion theorems, and isotropy.
• Quantitative Helly-type inequalities for log-concave families.
• Constructive algorithms for decomposition of the identity in measure-theoretic and numerical forms.

This body of work places the functional and high-dimensional theory of the John ellipsoid at the scientific interface of convex geometry, variational analysis, and computational mathematics, providing new tools for high-dimensional convex analysis and measure-theoretic applications (Baêta, 2 Apr 2025, Ivanov et al., 2020).