A new proof of Vazsonyis conjecture

We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n-2. The proof avoids the ball polytopes used in the original proofs by Grunbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane.

💡 Research Summary

The paper presents a self‑contained combinatorial proof of Vazsony’s conjecture, which asserts that for any set of (n) points in Euclidean three‑space the number of diameter pairs (pairs of points at the maximal mutual distance) does not exceed (2n-2). The authors deliberately avoid the classical approach that relies on ball polytopes—a sophisticated geometric construction introduced by Grunbaum, Heppes, and Straszewicz. Instead, they develop a proof that uses only elementary spherical geometry, planar graph theory, and Euler’s formula, thereby offering a more transparent and potentially extensible argument.

The exposition begins with precise definitions. A diameter pair is a pair of points whose distance equals the diameter of the whole set. The diameter graph has the points as vertices and an edge for each diameter pair. Known properties of such graphs are recalled: they are 2‑connected, every vertex has degree at least two, and the conjectured upper bound on the number of edges is (2n-2).

The central construction introduces a diameter region (R(x)) on the unit sphere for each point (x). (R(x)) consists of the spherical cap bounded by the great‑circle arcs that join (x) to each of its diameter partners. A crucial observation is that for distinct points (x) and (y) the regions (R(x)) and (R(y)) are interior‑disjoint; they may meet only along common boundary arcs, which correspond exactly to edges of the diameter graph. Consequently the collection ({R(x)}) yields a spherical triangulation of the sphere: each region is a spherical polygon whose edges are in one‑to‑one correspondence with the incident diameter edges.

Because each vertex of the diameter graph has degree at least two, every region (R(x)) contains at least two boundary arcs. Using elementary spherical geometry one can bound the area of each region from below by a constant that depends only on the minimal angular separation of the corresponding diameter edges. Summing over all (n) regions gives a total area that cannot exceed the surface area of the sphere, (4\pi). This area inequality translates directly into an inequality on the number of edges (E) of the diameter graph.

The authors then apply Euler’s formula to the spherical triangulation: (V - E + F = 2), where (V=n) and (F) is the number of spherical faces. By substituting the area bound and the fact that each face is at least a triangle, they derive the sharp inequality (E \le 2n-2). The proof is constructive: it shows that any configuration attaining the bound must correspond to a triangulation where every region is a spherical triangle, which in turn forces the original point set to lie on a convex polyhedron with a very specific combinatorial structure.

A notable corollary follows immediately. By projecting each spherical region (R(x)) onto the real projective plane (\mathbb{RP}^2) and preserving the adjacency relations, the diameter graph can be embedded in (\mathbb{RP}^2) without edge crossings. This provides a direct, combinatorial proof of the classical result that every three‑dimensional diameter graph is projective‑planar, a fact previously obtained via more topological arguments.

The paper concludes with a discussion of potential extensions. The authors argue that the same “region‑based” approach could be adapted to higher dimensions, where one would replace spherical caps by appropriate spherical polytopes and use higher‑dimensional analogues of Euler’s characteristic. They also suggest that the method may shed light on related problems such as the structure of diameter graphs in normed spaces, the existence of diameter triangles, and coloring properties of diameter graphs.

In summary, the work delivers a clean, elementary proof of the (2n-2) upper bound for diameter pairs in (\mathbb{R}^3), eliminates the need for ball‑polytope machinery, and simultaneously provides a constructive embedding of any three‑dimensional diameter graph into the projective plane. The approach not only clarifies the underlying combinatorial geometry but also opens avenues for further research in higher dimensions and in more general metric settings.