Paul Levy’s cyclic-polygon ratio conjecture

Determine whether, for every cyclic n-gon with side lengths a1, a2, ..., an, area An, and pseudo-area Pn defined by Pn = (Ln²/4)√∏i(1 − 2ai/Ln), the ratio ϕn = An/Pn satisfies ϕ⁰n < ϕn < 1, where ϕ⁰n is the ratio for the regular n-gon.

Background

The paper studies isoperimetric inequalities for plane n-gons inscribed in a circle. It compares the area An of a cyclic polygon with Paul Levy’s pseudo-area Pn, which depends on the polygon’s perimeter and side lengths. Equality is already known for triangles and quadrilaterals.

Paul Levy’s conjecture asserts that, for every cyclic n-gon, the area-to-pseudo-area ratio lies strictly between the corresponding ratio for the regular n-gon and 1. The paper proves the conjecture for polygons sufficiently close to regular polygons and for several special deformation families, but does not establish it for arbitrary cyclic polygons.