Double Sphere Property

• Double Sphere Property is a geometric condition in Banach spaces that ensures every convex combination of slices of the unit ball intersects the unit sphere.
• It characterizes diameter richness by guaranteeing the presence of two points at maximal distance in convex subsets, closely linking to the Strong Diameter Two Property.
• In discrete geometry, the concept extends to double-normal pairs on spheres, providing quantitative bounds and insights into symmetric extremal configurations.

The Double Sphere Property is a term that arises in several distinct mathematical contexts, notably in Banach space theory (as a property of the geometry of convex bodies in normed spaces) and in convex or discrete geometry on the sphere. The property is also indirectly connected to geometric characterizations of balls in spaces of constant curvature, where the intersection or convex-hull symmetry under congruences plays a central role. This article systematically presents precise definitions, major results, and related phenomena as developed in foundational works such as López-Pérez–Martín–Rueda Zoca (Lopez-Perez et al., 2017), Pach–Swanepoel (Pach et al., 2014), and Jerónimo-Castro–Makai (Jerónimo-Castro et al., 2024).

1. Definition of the Double Sphere Property in Banach Spaces

Let $X$ be a real Banach space with closed unit ball $B_X = \{x\in X : \|x\|\le 1\}$ and unit sphere $S_X = \{x\in X : \|x\|=1\}$.

Slice. Given $f\in S_{X^*}$ (the unit dual sphere) and $\alpha>0$, a slice of $B_X$ is $$S(B_X, f, \alpha) = \{\,x\in B_X : f(x)>1-\alpha\,\}$$.

Convex Combination of Slices. Any set of the form $C = \sum_{i=1}^n \lambda_i S(B_X, f_i, \alpha_i)$, with $\lambda_i\ge0$ and $\sum_{i=1}^n \lambda_i = 1$.

Double Sphere Property (Property (CS)). The Banach space $X$ is said to have the Double Sphere Property if every non-empty convex combination of slices of $B_X$ intersects the unit sphere: $$\forall\,C=\sum_{i=1}^n\lambda_i S(B_X, f_i, \alpha_i),\quad C\cap S_X\neq\emptyset.$$ Equivalently, by Theorem 3.4 in (Lopez-Perez et al., 2017), $X$ has the Double Sphere Property if and only if every such $C$ contains two points at distance exactly $2$: $\forall\,C,\,\exists\,x,y\in C: \|x-y\|=2.$ The property can be interpreted as a quantitative measure of the “diameter richness" of the geometry of the unit ball and is closely related to the Strong Diameter Two Property (SD2P): $\operatorname{diam}(C)=2$ for all convex combinations of slices.

2. Double-Normal Pairs and Spherical Double-Normal Property

In discrete and convex geometry, the notion of double-normal pairs provides a spherical analogue.

Given a finite set $S\subset S^2$ (the unit sphere in $\mathbb{R}^3$), a pair $\{p,q\}$ is called a double-normal pair if $S$ lies in the closed strip between the planes through $p$ and $q$ perpendicular to the chord $pq$ (i.e., for all $r\in S$, %%%%24%%%% and $\langle r-q,\, p-q\rangle \le 0$). In geometric terms, this means $S$ lies within the closed spherical strip bounded by specific great circles tangent at $p$ and $q$.

Such double-normal configurations provide a quantitative descriptor of how “spread out” a point set is on the sphere, and the count and structure of double-normal pairs serve as invariants analogous to the Double Sphere Property in infinite-dimensional settings (Pach et al., 2014).

3. Main Theorems: Equivalences and Bounds

Banach Space Characterizations

Theorem 3.4 in (Lopez-Perez et al., 2017) establishes:

• $X$ has the Double Sphere Property if and only if every convex combination of slices of $B_X$ contains two points at norm-distance exactly $2$.
• SD2P (diameter two property) is strictly weaker: every convex combination of slices has diameter $2$, but not all contain pairs with norm $1$.

Logical relations among various sphere/double-slice properties are captured in the following implication diagram: $$\begin{array}{ccccccc} \text{(W1)} &\implies& \text{(N1)} &\implies& \text{(W2)} & & \ &&\downarrow&&\downarrow&& \ &&\text{(N2)}&\implies&\text{(CS)} & & \end{array}$$ (See Section 4 for property definitions.) None of the converses generally holds.

Spherical Discrete Geometry: Quantitative Bounds

For finite $S\subset S^2$, Pach–Swanepoel (Pach et al., 2014) prove: $$N(S) \le \frac{17}{4}n-6 \quad \text{for} \quad n\ge8$$ where $N(S)$ is the total number of double-normal pairs. The bound is sharp, with equality if and only if $S$ is centrally symmetric and $\operatorname{conv} S$ is a polytope all of whose faces are rectangles or acute triangles (with exactly three rectangles at each vertex).

For strict double-normal pairs ($N'(S)$, with no other points on the bounding planes), the bound is: $N'(S) \le 2n-2$ Again, sharp for all $n\ge4$.

In the Euclidean planar case, the extremal value is $3\lfloor n/2\rfloor$ for $n\ge3$, with the extremal configuration being a centrally symmetric convex $n$-gon.

4. Relationship to Convexity, Weak Openness, and Topology

The Double Sphere Property is distinct from topological properties of convex combinations of slices:

• (N1): relative norm openness,
• (N2): non-empty relative norm interior,
• (W1): relative weak openness,
• (W2): non-empty relative weak interior, and their weak*-analogues.

Every Banach space has (N2), but (CS) can fail (e.g., strictly convex spaces satisfy (N1) but not (CS)). Weak*-openness and non-empty weak*-interior in $L_\infty$-spaces are equivalent to the underlying measure being purely atomic. For $C(K)$-spaces, the presence of atom-free measures implies failure of (W2).

This structural variability underscores that the Double Sphere Property is a more geometric or “diametric” phenomenon than a topological or convex-analytic one (Lopez-Perez et al., 2017).

5. Geometric and Metric Characterizations in Spaces of Constant Curvature

The Double Sphere Property appears implicitly in geometric characterizations of bodies in $S^d$, $\mathbb{R}^d$, $H^d$ whose pairwise intersections (or convex hulls of congruent copies) are always centrally symmetric.

Jerónimo-Castro and Makai (Jerónimo-Castro et al., 2024) prove:

• In $S^d$, $\mathbb{R}^d$, or $H^d$, if for all congruences $g,h$ the intersection $$\operatorname{int}(gK\cap hL)\neq\emptyset \implies gK\cap hL$$ is centrally symmetric, then $K,L$ are congruent spheres (and, in hyperbolic space, possibly paraspheres/horospheres).
• Analogously, if every convex hull of two congruent copies is centrally symmetric, then $K,L$ are congruent balls.

These characterize spheres as the unique bodies in such spaces exhibiting the "double-sphere" symmetry property under congruences.

6. Extremal and Illustrative Configurations

Notable explicit examples in the double-normal context:

• The regular cube's eight vertices inscribed in $S^2$: $N = 28 = (17/4) \cdot 8 - 6$
• Small rhombicuboctahedron ($n=24$ vertices): $N = 90 = (17/4) \cdot 24 - 6$
• For large $n$, arrangements of points along latitude circles and their antipodes, achieving $N = (17/4)n - O(\sqrt{n})$, with exact attainment infinitely often.

7. Context and Broader Implications

The Double Sphere Property and its geometric analogues serve as probes of the interplay between convexity, symmetry, and extremal configurations. In Banach space theory, the property demarcates spaces in which the “diameter two” phenomenon is witnessed by pairs on the unit sphere, reflecting underlying facial and extremal structure. In discrete and convex geometry, analogous double-normal constructs reveal combinatorial constraints and symmetry-induced extremality on the sphere, and similar symmetry conditions characterize metric balls in constant-curvature geometries.

The precise interplay of these properties with topological, measure-theoretic, and combinatorial invariants continues to motivate research across convex analysis, the geometry of Banach spaces, and metric geometry (Lopez-Perez et al., 2017, Pach et al., 2014, Jerónimo-Castro et al., 2024).