Aran S. Ziegler

A proof of Bezdek's conjecture on section symmetry

We prove Bezdek’s 1999 conjecture on Euclidean reflection symmetry of planar sections. A compact convex body with nonempty interior in three-dimensional Euclidean space, whose every affine plane section admits an orthogonal reflection, is an ellipsoid or a Euclidean body of revol...

A counterexample to the complex polynomial Koebe radius conjecture

We disprove the complex-coefficient polynomial Koebe radius conjecture proposed by Dmitrishin, Smorodin and Stokolos in 2018. We give a normalised cubic with rational real and imaginary coefficients that is univalent on a neighbourhood of the closed unit disk and omits a point of...

All even logarithmic differences of Boros–Moll coefficients

We prove a conjecture posed by Chen and Gu in 2008 by showing that every admissible even-order difference of the logarithms of a Boros–Moll coefficient row is strictly negative. The fourth-order case gives the conjectured ratio log-concavity, including the endpoint ratios. We est...

Off-line Ehrhart roots for complete multipartite symmetric edge polytopes

We disprove the 2016 conjecture of Higashitani, Kummer and Michałek that the Ehrhart polynomial of every complete multipartite symmetric edge polytope has all its roots on the line Re z = −1/2. The graph with six parts of size two gives an eleven-dimensional example with exactly...

Log-concavity of the Chen–Gu ratio sequence of Boros–Moll coefficients

In this preprint, we prove Chen and Gu’s 2008 conjecture on log-concavity of the ratio sequence of Boros–Moll coefficients, with strict inequality at every interior index. A positive integral interpolation and explicit fourth-derivative bounds establish the result for m ≥ 200. Th...