Origin-Symmetric Star Bodies

• Origin-symmetric star bodies are centrally symmetric, compact sets in ℝⁿ defined by continuous Minkowski gauges and unique ray intersections.
• They are characterized by symmetry detection via conical-section functions and hyperplane section analyses, ensuring structural rigidity.
• Integral geometric methods, including the Funk transform, enable explicit reconstruction of these bodies, linking convex geometry with functional analysis.

An origin-symmetric star body is a central object in convex and geometric analysis, particularly in the study of convex bodies and their generalizations. A star body in $\mathbb{R}^n$ is a compact set containing the origin in its interior, such that every ray from the origin meets the boundary exactly once, with a continuous Minkowski gauge or radial function. Origin-symmetric star bodies, the focus of this entry, are star bodies invariant under reflection through the origin; that is, $K = -K$. These objects encode both rich analytic structure and deep symmetry constraints, linking convexity, functional analysis, and integral geometry.

1. Definitions and Foundational Properties

A compact set $K \subset \mathbb{R}^n$ is a star body (with respect to the origin) if:

• $0 \in \operatorname{int}(K)$,
• For every $u \in S^{n-1}$, the ray $\{ t u : t \geq 0\}$ meets $\partial K$ at exactly one nonzero $r$,
• The Minkowski gauge

$$\|x\|_K := \inf\{ \lambda \geq 0 : x \in \lambda K \}$$

is continuous on $\mathbb{R}^n$.

The radial function $\rho_K : S^{n-1} \to (0, \infty)$ is given by

$$\rho_K(u) = \sup \{ r \geq 0 : ru \in K \} = \frac{1}{\|u\|_K}.$$

$K$ is origin-symmetric (centrally symmetric about 0) if $K = -K$, equivalently $\rho_K(\theta) = \rho_K(-\theta)$ for all $\theta \in S^{n-1}$ (Zhang, 4 Dec 2025Myroshnychenko et al., 2016Zawalski, 2023).

In the context of star-convex bodies, the requirement is that $[0, x] \subset K$ for all $x \in K$. Convex bodies are special cases of star bodies, but much of the theory extends to star-shaped (non-convex) sets provided the radial function is continuous and positive everywhere.

2. Symmetry Detection and Characterization

One of the pivotal analytic features of origin-symmetry in star bodies is the detection of this property by integrals of the radial function. Specifically, for a $C^1$ star body, consider the conical-section function

$$C_K(\xi,z) := \mathrm{vol}_{n-1}(K \cap C(\xi, z)),$$

where

$$C(\xi,z) = \{0\} \cup \{ x \in \mathbb{R}^n \setminus \{0\} : \langle x/|x|, \xi \rangle = z \}.$$

The Ryabogin–Yaskin symmetry detection theorem asserts that $K$ is centrally symmetric if and only if, for every $\xi \in S^{n-1}$,

$\frac{d}{dz} C_K(\xi,z) \Big|_{z=0} = 0,$

i.e., the one-sided derivatives of all conical-section functions vanish at the origin (Jr. et al., 2014). This criterion, and its roots in the Makai–Martini–Ódor theorem, formalizes the analytic detectability of central symmetry via sphere-valued data of the radial function.

3. Structural Rigidity: Characterization via Sections

The most powerful structural theorems for origin-symmetric star bodies employ information about their sections. Zhang proved that if $K \subset \mathbb{R}^n$ is an origin-symmetric star body ($n \geq 3$) and every hyperplane section $K \cap \xi^\perp$ is a centered ellipsoid, then $K$ itself is an ellipsoid: $$\text{If }\forall \xi \in S^{n-1},\; K \cap \xi^\perp \text{ is a centered ellipsoid,} \;\to\; K\text{ is a (centered) ellipsoid}.$$ The method amplifies symmetry via analysis of maximal and minimal radial directions,

$$M(K) = \{ \theta \in S^{n-1} : \rho_K(\theta) \geq \rho_K(\eta)\;\forall\eta \},\;\; m(K) = \{ \theta \in S^{n-1} : \rho_K(\theta) \leq \rho_K(\eta)\;\forall\eta \},$$

and iteratively applies diagonal linear maps to "equalize" axes via a process resembling axis symmetrization, culminating in the conclusion that $K$ is an ellipsoid (Zhang, 4 Dec 2025).

Further rigidity emerges from symmetry in sections—if every hyperplane section is completely symmetric, where completeness means any ellipsoid with the same symmetry group is a Euclidean ball, then the star body is a ball. In the case where each section is both origin-symmetric and 1-symmetric (invariant under the cube symmetry group), then the body must be a Euclidean ball (Myroshnychenko et al., 2016).

For bodies where every hyperplane section is a body of affine revolution (i.e., admits an affine axis of revolution and the action of $\mathrm{O}(n-1)$ on the orthogonal subspace), origin-symmetric star-convexity with sufficient smoothness ($C^3$ boundary) ensures that the whole body is also a body of affine revolution, provided $n \geq 4$ (Zawalski, 2023).

Sectional Characterization Table

Section property	Implies body is	Reference
Ellipsoidal sections (all hyperplanes)	Ellipsoid	(Zhang, 4 Dec 2025)
All sections completely symmetric	Euclidean ball	(Myroshnychenko et al., 2016)
All sections 1-symmetric/centered	Euclidean ball	(Myroshnychenko et al., 2016)
Affine revolution sections (with $C^3$)	Body of affine revolution ($n \geq 4$)	(Zawalski, 2023)

4. Integral Geometry and Reconstruction

Origin-symmetric star bodies are classically reconstructed from section data via integral transforms. Rubin established explicit reconstruction from central half-section volumes using hemispherical Radon-type transforms and the Funk transform on the sphere: $$(R^+_k f)(\xi) = \int_{\xi_+} f(\theta)\, d_{\xi}\theta,\quad v^+(\xi) = \frac{1}{k} (R^+_k(\rho_K^k))(\xi),$$ where $\xi$ runs over $k$-dimensional subspaces, and $f$ is taken to be $\rho_K^k$. The evenness imposed by origin-symmetry is essential—the classic Funk transform annihilates odd functions. A star body's (even) radial function is recovered from the section data via an explicit inversion formula: $$\rho_K(\theta)^k = 2\, [F_k^{-1}(k v^+)](\theta),\quad \theta_n > 0,$$ where $F_k^{-1}$ denotes any known inversion of the Funk transform (Rubin, 2016).

This process yields an explicit, linear, and constructive algorithm for determining origin-symmetric star bodies from partial volume data, with precise minimal data sets calculable via dimension counting.

5. Symmetry Groups, Complete Symmetry, and Isotropy

The symmetry group of a set $A \subset \mathbb{R}^n$ is $\Sym(A) = \{ T \in \mathrm{ISO}(n) : T(A) = A \}$; for a function on $S^{n-1}$, $\Sym(f) = \{ T \in \mathrm{O}(n) : f(Tx) = f(x) \;\forall x \}$. A subgroup $G \subset \mathrm{ISO}(n)$ is complete if any ellipsoid with $G$ in its symmetry group is a Euclidean ball.

A star body $K$ is "completely symmetric" if its centroid is at the origin and its symmetry group is complete. Symmetry assumptions on sections, combined with the isotropy of restrictions of the radial function to equators, yield classification results: if almost all equators restrict to isotropic functions, then $K$ must be a ball (Myroshnychenko et al., 2016).

The isotropy condition for a function $f$ on the sphere involves both center of mass vanishing and a uniform quadratic moment: $\int_{S^{n-1}} x\, f(x)\, d\sigma(x) = 0,$

$$\int_{S^{n-1}}(x \cdot y)^2 f(y)\, d\sigma(y) = \lambda,$$

for constant $\lambda > 0$ and all $x \in S^{n-1}$.

6. Applications, Examples, and Open Problems

Applications range from finite-dimensional Banach space theory (notably, the Banach isometric-subspace problem; see (Zhang, 4 Dec 2025)) to geometric tomography and convex geometry, especially in rigidity and stability results. A notable application is the characterization of ellipsoids as the only convex bodies for which all maximal-volume inscribed ellipsoids in sections coincide under affine transformations.

Key examples distinguish the sharpness of these results:

• In $\mathbb{R}^3$, cubes have planar sections which are centrally symmetric (squares or regular hexagons), but the cube itself is not a body of revolution. This highlights the necessity of dimension or additional hypotheses in certain structure theorems (Zawalski, 2023).
• It is possible for an origin-symmetric convex body to have all projections unconditional, yet not be an ellipsoid or body of revolution (Myroshnychenko et al., 2016).

Persistent open questions include the determination of bodies from projections subject to various symmetry constraints, as well as local versions of isotropy-induced constancy. For example, whether all projections being affine images of completely symmetric sets implies that the body is an origin-symmetric ellipsoid remains unresolved (Myroshnychenko et al., 2016).

7. Impact in Convex Geometry and Functional Analysis

Origin-symmetric star bodies form the analytic and geometric basis for multiple characterizations and rigidity phenomena in high-dimensional convex geometry. They bridge convex analysis, isotropic measures, and integral-geometric transforms, serving as test objects for symmetry detection, normed space structure, and affine classification. Their centrality is evident in major advances such as the resolution of Banach's isometric-subspace problem (Zhang, 4 Dec 2025), explicit reconstructions in tomography (Rubin, 2016), and symmetry-based criteria for uniqueness and stability (1411.44801611.09443).

The study of origin-symmetric star bodies remains a fertile intersection of geometry, analysis, and algebraic symmetry, with ongoing research probing the delicate interactions between local section data, global geometry, and symmetry constraints.