Vertex $\pi$-Floodlights

Definitions and mathematical statement

Let $P\subset\mathbb R^2$ be a simple polygonal region with $n\geq3$ vertices. A $\pi$-floodlight at a vertex $v$ has aperture $\pi$, represented by a halfplane $H_v$ whose boundary passes through $v$. Under the usual visibility model it illuminates \[ I(P;v,H_v)=\{x\in P\cap H_v:[v,x]\subseteq P\}. \] Choose at most one floodlight at each vertex, with an admissible orientation for each chosen light, so that the union of illuminated sets equals $P$. If $g(P)$ is the minimum number needed under the specified orientation convention, determine the worst-case function \[ F(n)=\sup\{g(P):P\text{ is a polygon with }n\text{ vertices}\}. \] The question seeks a bound valid for every such polygon, with sharp worst cases.

The source specifies "inward-facing" lights and says their defining halfspace lies inside the polygon locally near the vertex. A literal inclusion would be $H_v\cap B_\varepsilon(v)\subseteq P$ for some $\varepsilon>0$. Such a halfplane cannot fit into a convex corner of angle less than $\pi$, so this literal convention would leave a strictly convex polygon with no admissible vertex lights. It therefore needs clarification before the stated all-polygons bound can be formalized. Reversing this inclusion or allowing general orientations would define different models and is not silently done here.

The maintained TOPP entry, as reported in the record, says the problem is open and that the sharp universally sufficient asymptotic fraction lies between $5/8$ and $2/3$. Exact integer rounding and additive terms are not supplied. The source distinguishes the intended inward-facing model from models with arbitrary orientations or restricted edge-aligned orientations.

Short English statement

Find the sharp worst-case number of vertex lights with aperture $\pi$ needed to illuminate an $n$-vertex polygon, using at most one light per vertex. The intended local inward-facing convention must be clarified before applying the reported $5/8$--$2/3$ bounds.

Sources

• The Open Problems Project, Problem 23, "Vertex $\pi$-Floodlights," https://topp.openproblem.net/p23; accessed 2026-07-29, source-TeX extraction. The maintained entry is reported as open, with the universally sufficient fraction between $5/8$ and $2/3$.
• Public TOPP source repository, https://github.com/edemaine/topp. The supplied rights note says its LICENSE expressly scopes software, leaving problem-text reuse NEEDS_REVIEW.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $5500023$, source code catalogue entry, Vertex $\pi$-Floodlights. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.