Rademacher's Half-Ball and Cone Counterexample in Three Dimensions
Manuscript 13 October 2026 · Online 13 October 2026
Abstract
Rademacher proposed a half-ball joined to a cone, with a small tip truncation, as a counterexample to monotonicity of the expected volume of a random tetrahedron under inclusion. We prove that this specific construction works in dimension three, with cone height \(10^{-5}\) and truncation height \(10^{-10}\). An integer-only outward interval certificate proves that the half-ball expectation exceeds its origin-anchored counterpart \(9\pi/1024\) by more than \(1/6000\). A quantitative mixture argument transfers this gap to explicit nested convex bodies.
General three-dimensional nonmonotonicity was already proved by Kunis, Reichenwallner and Reitzner. This AI-assisted, computer-assisted, self-audited and unrefereed preprint verifies the specific half-ball construction. No exact closed-form expectation, absolute priority, independent review or formal verification is claimed.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete construction-specific counterexample
- Categories
- math.PR · math.MG
- Manuscript
- 13 October 2026
- Online release
- 13 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Rademacher's Half-Ball and Cone Counterexample in Three Dimensions,” EulerSolve Research Papers, OWR-4136-005, 2026. https://doi.org/10.5281/zenodo.23350280.
BibTeX
@misc{ferudun2026halfball,
author = {Ferudun, Alper},
title = {Rademacher's Half-Ball and Cone Counterexample in Three Dimensions},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-4136-005/},
doi = {10.5281/zenodo.23350280},
note = {OWR-4136-005; unrefereed preprint}
}