# Does the smallest enclosing copy fit? Random points in a convex polygon

Open problem settled, formally verified in Lean 4. Vamshi Jandhyala.

> For many random points in an equilateral triangle, the smallest enclosing copy fits inside with probability tending to 13/48, not 1/2.

Canonical: https://vamshij.com/research/smallest-enclosing-copy
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/smallest-enclosing-copy
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/smallest-enclosing-copy/paper/enclosing.pdf

Drop `n` random points into a disk and draw the smallest circle containing them: does it lie inside the disk? An answer on Mathematics Stack Exchange shows that the limiting probability is 1/2 ([question 4799757](https://math.stackexchange.com/q/4799757)). A follow-up on MathOverflow asked whether the same holds for a regular polygon and a smallest enclosing regular polygon, which may be rotated ([question 458571](https://mathoverflow.net/q/458571)).

The answer is no. For `n` uniform points in a convex polygon `K`, the probability that some smallest similar copy of `K` containing them lies inside `K` converges to an explicit constant `p(K)`, given by a formula over a Poisson limit model (Theorems 1 and 10). For triangles the formula is explicit (Corollary 2): `p = 13/48` for the equilateral triangle, `7/24` for the right isosceles triangle, `29/96` for the 30-60-90 triangle. For the regular `q`-gon (Corollary 3),

`p_q = q tan(π/q) E[Vol(T) 1{0 ∈ int T}] + 1{q even} 8/(3q²)`,

where `T` is the tetrahedron spanned by four independent points, each on a uniformly chosen vertical edge of the prism `P_q × [-1, 1]` at a uniform height; `p_4 = 1/4`, and numerically `p_q` settles near 0.146, far from the disk's 1/2. The reason is the freedom to rotate the enclosing copy, which the disk does not have.

Preprint v1, 8 October 2026, not peer reviewed. Every theorem, lemma and corollary is formally verified in Lean 4 (`lean/`, standard axioms only); the exposition has not yet been independently reviewed. The author used AI tools (Claude, Anthropic; Codex, OpenAI) in this work, as described in the paper's acknowledgement, and is responsible for its content.