# Catching the centroid

Partial progress on an open problem. Vamshi Jandhyala.

> A circle on a random diameter of a convex region contains the centroid with probability at most 14/27 ≈ 0.5185; the equilateral triangle's 0.5164 is conjectured best.

Canonical: https://vamshij.com/research/catching-the-centroid
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/catching-the-centroid
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/catching-the-centroid/paper/note.pdf

Pick two random points `X` and `Y` in a planar convex region `K` and draw the circle with diameter `XY`. Dan asked on Mathematics Stack Exchange ([question 5101873](https://math.stackexchange.com/q/5101873)) which convex shape makes this circle most likely to contain the centroid. A disc or a square gives exactly `1/2` and an equilateral triangle gives `1/3 + ln 3 / 6 ≈ 0.5164`; the answers there conjecture that the equilateral triangle is best. That conjecture is still open.

The note shows that the excess over `1/2` comes only from the asymmetry of `K` about its centroid. In polar coordinates about the centroid, `P(K) − 1/2` is the integral of the odd part of the angular density against the imbalance of the half-planes (Proposition 1), equivalently half the area swept by the imbalances of two perpendicular lines. Bounding it by the product of Grünbaum's half-plane asymmetry and the asymmetric share `1 − s(K)` of the region (Proposition 2), and using Stewart's 1958 theorem that `|K ∩ (2G − K)| ≥ (2/3)|K|` at the centroid `G`, gives:

- **Theorem 1.** `|P(K) − 1/2| ≤ 1/54` for every convex region, so `P(K) ≤ 14/27 ≈ 0.5185`, within `0.0021` of the conjectured value.

The note also gives a three-line proof of Stewart's inequality from Grünbaum's for regions whose boundary crosses its reflection through the centroid six times, which every region close to a triangle does (Lemma 1, Proposition 3), credits Levi's older bound `2/5`, and explains why `14/27` cannot reach `0.5164`: both factors of Proposition 2 are affine invariants, while `P` is not.

Preprint v1, 10 October 2026, not peer reviewed and not yet independently reviewed. Propositions 1, 2 and 3, formula (3), Lemma 1, Theorem 1 and the bound `8/15` are formally verified in Lean 4 (`lean/`, standard axioms only) for continuous periodic angular densities; Grünbaum's, Stewart's and the Minkowski–Radon inequalities are quoted and enter as hypotheses, and the polar-coordinate formula (1) is the Lean definition of `P`. The author used AI tools (Claude, Anthropic) in this work, as described in the paper's acknowledgement, and is responsible for its content.