Characterization of the sphere by means of congruent support cones

Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this work we are going to present a result which claims that if for every two support cones $C_x$, $C_y$ of $M$, with apexes $x,y \in K$, respectively, there exists $Φ$ in the semi direct product of the orthogonal group $O(3)$ and $\mathbb{E}^3$ such that $$C_y=Φ(C_x),$$ and this can be done in a continuous way, then $M$ is a sphere.

💡 Research Summary

The paper investigates a geometric rigidity problem in three‑dimensional Euclidean space. Let M be a convex body and K a closed convex surface that encloses M (M ⊂ int K). For each point x on the boundary of K, one can form the support cone Cₓ of M with apex x: the cone generated by all rays from x through points of M. The central hypothesis is twofold: (i) any two such cones Cₓ and C_y are affinely congruent, i.e. there exists an element Φ = a + Ω of the affine group A(3)=ℝ³⋊O(3) (a translation a and an orthogonal matrix Ω) with Φ(Cₓ)=C_y; and (ii) the choice of Φ can be made continuously as a function of the apex point on bd K.

The authors first formalize this continuity: for a convergent sequence of apexes xₙ→x, the corresponding orthogonal parts Ωₙ converge to Ω and the translated cones converge in the Hausdorff metric. They then exploit the topology of the principal bundle τ:S²→O(3) (the quotient map sending an orthogonal matrix to its action on a fixed unit vector). Since τ has no global section (the bundle is non‑trivial), one can find a point x*∈bd K and two sequences approaching x* that induce two distinct limiting orthogonal maps Ω_u* and Ω̄_u*. The composition Ω_u*⁻¹∘Ω̄_u* is either the identity or a non‑trivial symmetry of the reference cone Cₓ₀. Consequently each support cone Cₓ possesses either a plane of symmetry or an axis of symmetry.

If a cone has a plane of symmetry, either it has infinitely many such planes—in which case it must be a right circular cone—or it has only finitely many, which forces the existence of an axis of symmetry. Thus every support cone is either axis‑symmetric or a right circular cone.

Assuming the presence of an axis of symmetry Lₓ for each cone, the authors construct, for each x, a plane Δₓ orthogonal to Lₓ at unit distance from the apex, and define M_η(x)=Cₓ∩Δₓ. The map η:bd K→S² sending x to the unit vector parallel to Lₓ is shown to be a continuous bijection, hence a homeomorphism (using homology arguments). Therefore the family {M_η(x)} forms a continuous field of congruent convex bodies tangent to the unit sphere. By a result of Mani (1975), such a field in three dimensions can exist only if each body is a circle. Hence each cross‑section M_η(x) is a circle, which forces every support cone Cₓ to be a right circular cone.

Finally, the authors invoke Matsuura’s theorem (1979): if all support cones of a convex body with respect to a surrounding convex surface are right circular cones, then the body itself must be a Euclidean ball. Consequently, under the stated affine‑congruence and continuity assumptions, the convex body M is necessarily a sphere.

The paper also surveys related literature on ellipsoidal support cones, central symmetry, and the Banach Isometric Conjecture, and highlights the novel use of fiber‑bundle topology to bridge geometric conditions with topological obstructions. The main contribution is a clean characterization: continuous affine congruence of all support cones with apexes on a surrounding convex surface characterizes the Euclidean sphere in ℝ³.