Opaque forest problem on finding opaque sets for various planar shapes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $C\subset\mathbb R^2$ be a convex body. A rectifiable barrier presentation is a finite or countable family $\Gamma=(\gamma_i)$ of rectifiable curves in the plane, with finite total cost $L(\Gamma)=\sum_i\operatorname{length}(\gamma_i)$, whose union $B_\Gamma=\bigcup_i\gamma_i$ intersects every straight line meeting $C$. The barrier may be disconnected and may extend outside $C$. Define $\ell(C)=\inf_\Gamma L(\Gamma)$ over all such barrier presentations.

Statement recorded in the supplied source.

Determine the shortest barrier length $\ell(C)$ and optimal barriers for given planar convex bodies $C$, in particular a square, a disk, and an equilateral triangle. [S2]

Short English statement

For a given planar convex shape, find the shortest set of curves that intercepts every line passing through it. The curves may be disconnected or lie outside the shape.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200053 (AMR-071-0053). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Adrian Dumitrescu, Minghui Jiang and János Pach, Opaque sets (2012 version). Sections 1–2, pp. 1–4. https://arxiv.org/pdf/1005.2218.