Max-Min Solid Angle Convex Polyhedron

• Max-min-solid-angle convex polyhedron is a convex shape chosen from a finite point set that maximizes the smallest solid angle at its vertices.
• The optimal convex polyhedron emerges by applying a decremental greedy algorithm that iteratively deletes hull vertices with the minimum solid angle, ensuring convexity and improved angular quality.
• The approach achieves an O(n²) time bound by maintaining the convex hull dynamically, integrating rigorous geometric analysis with efficient computational methods.

Searching arXiv for the specified paper and closely related solid-angle/polyhedron work. A max-min-solid-angle convex polyhedron is a convex polyhedron selected from a finite point set $S \subset \mathbb{R}^3$ so as to maximize the smallest solid angle over its vertices. In the formulation studied in "Decremental Greedy Polygons and Polyhedra Without Sharp Angles" (Eppstein, 6 Jul 2025), the admissible objects are first taken to be all non-self-intersecting polyhedral surfaces whose vertices are a subset of $S$, and the principal result is that an optimal solution always exists as a convex polyhedron and can be found in time $O(n^2)$, where $n=|S|$.

1. Formal definition and geometric quantity

For a finite point set $S \subset \mathbb{R}^3$, the optimization problem considers polyhedral surfaces whose vertices are drawn from a subset of $S$. If $v$ is a vertex of a convex polyhedron, its solid angle is the angle of the cone formed by the faces incident to $v$; equivalently, it is the area on the unit sphere cut out by the normal directions of the faces incident to $v$. The objective is the max-min criterion

$\max_{P}\; \min_{v\in V(P)} \Omega(v),$

where $S$0 ranges over admissible polyhedral surfaces or polyhedra with vertices in $S$1, $S$2 is the vertex set of $S$3, and $S$4 is the solid angle at $S$5 (Eppstein, 6 Jul 2025).

Two features of the formulation are structurally important. First, the surface is required to be non-self-intersecting. Second, the solution is allowed to use only a subset of the input points rather than all of them. The optimization target is therefore not a triangulation or hull reconstruction problem in the usual sense, but a bottleneck optimization over local angular defects, with the bottleneck defined by the smallest vertex solid angle.

The paper does not re-derive a spherical formula for $S$6, but it explicitly uses the standard notion of the solid angle interior to the surface as viewed from any point, or vertex, of the surface. This places the problem in the standard spherical representation of polyhedral vertex geometry, where local cone geometry is measured by spherical area.

2. Convexity as an emergent property of the optimum

A central theorem is that the optimum over non-self-intersecting polyhedral surfaces can always be realized by a convex polyhedron. The proof proceeds by taking an arbitrary optimal non-self-intersecting surface and replacing it by its convex hull. This operation can only possibly delete some vertices and improve or preserve the solid angles at the vertices that remain, so the convex hull is also optimal (Eppstein, 6 Jul 2025).

This establishes that convexity is not imposed as a prior constraint on the search space. Rather, convexity emerges from the extremal structure of the max-min objective. A common source of confusion is to treat the result as an optimization over convex hulls from the outset; the paper’s statement is stronger, because it begins with the larger class of non-self-intersecting polyhedral surfaces and then proves that no optimum is lost by passing to the convex hull.

The same pattern appears in the planar analogue studied in the same work: in $S$7D, a convex polygon maximizing the minimum interior angle is obtained by the same convex-hull reduction. This parallel is significant because it shows that the max-min criterion interacts with convexification in a monotone way in both dimensions.

3. Decremental greedy structure

The algorithmic solution is a direct geometric instance of a more general decremental greedy theorem for monotone bottleneck subset problems. In that abstraction, one has a ground set $S$8, a quality function $S$9 for each element $O(n^2)$0 in a subset $O(n^2)$1, and a bottleneck objective

$O(n^2)$2

subject to the monotonicity condition

$O(n^2)$3

The theorem states that the decremental greedy algorithm always finds the unique maximal optimal subset (Eppstein, 6 Jul 2025).

For the polyhedron problem, the quality of a point is its solid angle when it is a vertex of the current convex hull. If a point is not currently on the hull, it is treated as better than any hull vertex in the bottleneck sense, because interior points do not determine the minimum solid angle. Under deletions, the quality of any remaining point can only stay the same or get worse: a point may become a hull vertex, and a hull vertex may lose neighbors and therefore have a smaller solid angle. This monotonicity is what makes greedy deletion safe.

The maintained state consists of the current set of not-yet-deleted points, its convex hull, the current sharpest solid angle on that hull, the best value seen so far, and the deletion prefix that achieved it. At each step, the algorithm deletes a vertex of the current convex hull with minimum solid angle, updates the best minimum-solid-angle value seen so far, maintains the convex hull of the remaining points, and at the end returns the convex hull obtained after removing the prefix of deletions that achieved the best value. In this sense, the desired polyhedron is not necessarily the final hull after all deletions; it is the best hull encountered during the decremental process.

The broader significance is methodological. The paper formalizes a class of monotone bottleneck subset problems for which decremental greedy is provably optimal, and the polyhedron problem is one of its geometric exemplars. The same framework also generalizes the max-min-angle polygon algorithm and a known algorithm for graph degeneracy.

4. Data structure and $O(n^2)$4 time bound

The stated running time for the three-dimensional problem is exactly $O(n^2)$5. This bound is obtained by explicitly maintaining the $O(n^2)$6D convex hull under deletions rather than by invoking a sophisticated dynamic hull data structure (Eppstein, 6 Jul 2025).

The decremental convex-hull structure uses a balanced binary search tree on the points ordered by $O(n^2)$7-coordinate. Each subtree stores the explicit convex hull of its point subset. After a deletion, the affected hulls are recomputed by merging child hulls. The amortized cost of one deletion is $O(n^2)$8, because the affected merges sum to a geometric series over subtree sizes. Since there are $O(n^2)$9 deletions, the total cost is

$n=|S|$0

The paper notes that more sophisticated decremental or fully dynamic hull structures exist, but the explicit method is sufficient both to establish the asymptotic bound and to keep the exposition self-contained. The computational contrast with the planar case is explicit: the analogous polygon problem is solvable in $n=|S|$1, whereas the $n=|S|$2D hull-maintenance burden raises the polyhedron bound to $n=|S|$3.

5. Relation to extremal solid-angle theory

The finite-point-set optimization problem is distinct from classical extremal questions that fix a combinatorial type or optimize over continuous families of polytopes. In "Bounding minimal solid angles of polytopes" (Akopyan et al., 2015), the guiding question is: among all simplices in $n=|S|$4, how large can the smallest solid angle be? For tetrahedra, the result is that every tetrahedron has a solid angle at some vertex no greater than that of the regular tetrahedron. Equivalently, in dimension $n=|S|$5, the regular tetrahedron realizes the conjectured upper benchmark for the minimum vertex solid angle.

That extremal simplex problem is also max-min in form, but it differs in two essential respects from the discrete extraction problem. First, it ranges over all simplices rather than over subsets of a prescribed point set. Second, its conclusions are comparison theorems relative to regular models, whereas the finite-point-set problem is an exact algorithmic optimization problem. A plausible implication is that both problems are governed by the same local quantity, $n=|S|$6, but they lie in different regimes: one continuous and comparison-based, the other discrete and algorithmic.

The same paper extends the discussion to polyhedra combinatorially equivalent to Platonic solids, proving that one solid angle and one dihedral angle are bounded above by the corresponding angle of the regular solid in $n=|S|$7 (Akopyan et al., 2015). This provides a broader extremal context for the local geometry being optimized in the decremental greedy construction.

A different line of work studies global integral functionals of visual solid angle. "On the solid angle of a convex set" (Bruna et al., 2023) defines, for an exterior point $n=|S|$8, a visual solid angle $n=|S|$9 and derives Crofton-type integral formulas involving $S \subset \mathbb{R}^3$0, $S \subset \mathbb{R}^3$1, and $S \subset \mathbb{R}^3$2. Its cleanest extremal statement is

$S \subset \mathbb{R}^3$3

with equality if and only if $S \subset \mathbb{R}^3$4 is a ball (Bruna et al., 2023). This is not a pointwise max-min theorem for polyhedra, but it shows that solid angle also supports global integral inequalities, with the sphere as the extremal body.

6. Computation of solid angles and higher-dimensional extensions

Because the decremental greedy algorithm depends on comparing vertex solid angles on successive convex hulls, solid-angle evaluation is a natural computational subproblem. "Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps" (Mazonka, 2012) develops formulas for the solid angle of conical surfaces and polyhedral cones. For a polyhedral cone with cyclic unit generators $S \subset \mathbb{R}^3$5, it gives the closed form

$S \subset \mathbb{R}^3$6

where $S \subset \mathbb{R}^3$7, $S \subset \mathbb{R}^3$8, $S \subset \mathbb{R}^3$9, and $S$0 are defined from consecutive scalar and triple products. In the tetrahedral or spherical triangle case, this reduces to the classical Van Oosterom–Strackee formula

$S$1

These formulas are directly relevant because the solid angle at a convex-polyhedron vertex is exactly the solid angle of the incident polyhedral cone (Mazonka, 2012).

In higher dimensions, "Solid angle measure of polyhedral cones" (Fitisone et al., 2023) studies normalized solid angles of polyhedral cones via multivariable hypergeometric series, together with decomposition methods that reduce arbitrary simplicial cones to pieces satisfying a positive-definite convergence criterion. The paper does not formulate or solve a max-min optimization problem over polyhedra, but it provides a computation-oriented toolkit for evaluating the local cone measures that would arise in higher-dimensional analogues (Fitisone et al., 2023).

Taken together, these works place the max-min-solid-angle convex polyhedron problem at the intersection of discrete geometric optimization, spherical cone geometry, and solid-angle computation. The 2025 result identifies a tractable $S$2D bottleneck optimization problem on finite point sets and shows that its optimum is realized by a convex polyhedron obtainable by decremental greedy deletion in $S$3 time (Eppstein, 6 Jul 2025).