Complex Ellipsoids: Geometry & Applications

• Complex ellipsoids are defined as the images of the unit ball under complex affine transformations, characterized by a positive-definite Hermitian quadratic form.
• They exhibit rich symmetry, rigidity, and invariant metrics that distinguish them from real ellipsoids, with implications in convex geometry and several complex variables.
• Applications include topological data analysis through ellipsoid complexes, which enhance persistence and classification accuracy in datasets with complex geometries.

A complex ellipsoid is the image of the unit ball under a complex affine transformation in $\mathbb{C}^n$, often realized as a subset of $\mathbb{C}^n$ defined by a positive-definite Hermitian quadratic inequality. These domains are central objects in convex geometry, several complex variables, and holomorphic function theory, and exhibit rich symmetry and rigidity properties not present in their real counterparts. This article surveys definitions, characterizations, geometric invariants, extremal problems, invariant metrics, and applications, referencing both foundational and recent research.

1. Definition, Matrix Representation, and Convexity

Formally, the standard complex ellipsoid in %%%%2%%%% is given by

$$E_{A,b} = \left\{z \in \mathbb{C}^n : (z-b)^* M (z-b) \leq 1 \right\}$$

where $M = (A A^*)^{-1}$ for invertible $A \in \mathbb{C}^{n \times n}$ and $b \in \mathbb{C}^n$. The matrix $M$ must be Hermitian positive-definite; this condition is necessary and sufficient for $E_{A,b}$ to be convex (Arocha et al., 2022).

In coordinates adapted to the diagonal form, every centered complex ellipsoid is

$$E=\left\{ z\in\mathbb{C}^n: \sum_{j=1}^n \lambda_j |z_j|^2 \leq 1 \right\}, \quad \lambda_j > 0$$

(Arocha et al., 2021). The extreme points of $E$ satisfy $(z-b)^* M (z-b) = 1$, forming a smooth, strictly convex hypersurface.

2. Geometric Characterizations and Complex Symmetry

Complex ellipsoids are rigidly characterized by the properties of their sections and projections. Specifically, an affine convex body $K\subset \mathbb{C}^n$ is a complex ellipsoid if and only if every complex line section $K\cap L$ is a (possibly degenerate) disk (“Bombon criterion”) (Arocha et al., 2021, Arocha et al., 2022). This is stricter than the real case: in $\mathbb{R}^n$, line–section–segment characterization is valid only under stronger regularity or dimension assumptions.

The notion of complex symmetry—$S^1$ invariance under $z\mapsto \xi z$ for $\xi \in S^1$—is central. A convex body is symmetric iff for every $\xi\in S^1$, $\xi K$ is a translate of $K$. Sections and projections inherit symmetry properties, and various theorems establish equivalence between symmetry of hyperplane sections, projections, and the global symmetry of the domain (Arocha et al., 2021, Arocha et al., 2022).

Characterization Table

Criterion	Implies Ellipsoid?	Reference
All line sections are disks	Yes	(Arocha et al., 2022)
All hyperplane sections symmetric	Yes	(Arocha et al., 2021)
All $k$-plane projections are ellipsoids	Yes	(Arocha et al., 2022)

3. Extremal Inscribed and Circumscribed Ellipsoids

Let $K\subset \mathbb{C}^n$ be a convex body. The minimal-volume circumscribed complex ellipsoid (MiCE) and maximal-volume inscribed complex ellipsoid (MaIE) are uniquely determined:

• If $E(a_1,A_1)$ and $E(a_2,A_2)$ are both maximal inscribed ellipsoids in $K$, then $A_2 = A_1$ and $a_2$ is a translate of $a_1$; uniqueness holds up to translation, strengthening the classical John–Löwner results (Arocha et al., 2020).
• The minimal circumscribed ellipsoid is unique, with both center and shape matrix determined (Arocha et al., 2020).

Additionally, the “Complex Brunn-type characterization” establishes that a “puck” (a set all of whose line slices are disks) is a complex ellipsoid iff the centers of all such slices are contained in an affine hyperplane. These properties depend fundamentally on the strict concavity of $A \mapsto \log\det(A)$ on Hermitian positive-definite matrices (Arocha et al., 2020).

4. Invariant Metrics and Curvature Structure

Complex ellipsoids are natural domains for several biholomorphically invariant metrics: the Carathéodory–Reiffen metric, the Bergman metric, and the complete Kähler–Einstein metric of negative scalar curvature. For

$$E = \left\{ z=(z_1,\dots,z_n)\in\mathbb{C}^n : \sum_{j=1}^n \frac{|z_j|^2}{a_j^2} < 1 \right\}$$

and $\sigma(z) = \sum_j \frac{|z_j|^2}{a_j^2}$, these metrics are: $$c_E(z; v) = \frac{\left( \sum_j \frac{|v_j|^2}{a_j^2} \right)^{1/2}}{1 - \sigma(z)}$$

$$B_E(z; v)^2 = (n+1)\frac{\sum_j \frac{|v_j|^2}{a_j^2}}{1-\sigma(z)} + (n+1)\frac{\left| \sum_j \frac{\bar{z}_j v_j}{a_j^2} \right|^2 }{(1-\sigma(z))^2}$$

$g_{KE,E}(z; v) = \frac{1}{n+1} B_E(z; v)$

These metrics are uniformly equivalent (i.e., differ by dimension-dependent constants), but not proportional unless $n=1$ (the disk) or the domain is the ball (Cho, 2018). For the 2-dimensional model $E_p = \{ (z,w) : |z|^2 + |w|^{2p} < 1 \}$, explicit curvature tensors and holomorphic sectional curvatures are computable, and metrics become proportional only for $p=1$ (disk case) (Cho, 2018).

5. Bergman Kernel Formulas for Ellipsoids and their Intersections

The Bergman kernel $K_D(z,w)$, a core invariant in several complex variables, admits closed formulas for complex ellipsoids and intersections thereof. For the Reinhardt domain $$D_{p,q,r} = \{ z \in \mathbb{C}^3 : |z_1|^p + |z_2|^q < 1, |z_1|^p + |z_3|^r < 1 \}$$, the kernel is expressible via multivariable hypergeometric functions (Lauricella–Appell $F_8$, $F_1$, $F_2$). For $p=q=r=2$, the explicit formula is: $$K_{D_1}(z, w) = \frac{3(1-v_1^2)(v_2 + v_3) - 2 v_1 (1-v_2 - v_3)^2}{\pi^3 (1-v_1-v_2)^3 (1-v_1-v_3)^3}$$ with $v_i = z_i \overline{w}_i$ (Beberok, 2015). Such formulas highlight the tractable analytic structure of complex ellipsoid domains. In the “disk-disk” case, $D_1$ is a Lu Qi–Keng domain: its Bergman kernel never vanishes (Beberok, 2015).

6. Applications in Topological Data Analysis: Ellipsoid Complexes

Recent advances leverage the geometry of complex ellipsoids to improve topological invariants for data applications. In “Persistent Homology via Ellipsoids,” tangent-aligned ellipsoid complexes derived from local PCA outperform classical Rips complexes for homology estimation and classification, especially in regimes with sparse sampling or narrow bottlenecks. The construction associates to each data point $x_i \in \mathbb{R}^D$ a local ellipsoid $E_i(r)$ with principal axes and scales determined by the local covariance matrix. Complexes built from ellipsoid intersections capture the geometry more faithfully than isotropic balls, leading to longer persistence intervals and enhanced classification accuracy (Kališnik et al., 21 Aug 2024).

7. Open Questions and Research Directions

Open problems include generalizations of “False-Center Theorems” to the complex case, characterization of complex-homothety under section equivalence, and extensions to infinite-dimensional settings, especially in Banach space theory. The “Complex Banach Conjecture” posits that finite-dimensional complex Banach spaces with all subspaces isometric must be Hilbert spaces, i.e., their unit balls are complex ellipsoids. Partial results hinge on topological obstructions and symmetry group reductions (Arocha et al., 2022).

Complex ellipsoids thus serve as a nexus for convex geometry, holomorphic analysis, metric geometry, and computational topology, with their rigidity, symmetry, and analytic tractability distinguishing them sharply from real ellipsoid analogues.