PreprintCC BY 4.0arXiv
Pinch's conjecture on a-convexity
Abstract
For a in ℝ, a subset V contained in ℝn is said to be a-convex if x, y ∈ V Longrightarrow a x + (1−a) y ∈ V. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the a-convex hull of V is the intersection of all of the a-convex subsets of ℝn that contain V, and Pinch also defines D(a) as the a-convex hull of { 0, 1 } in ℝ1. Pinch conjectured that if a is a totally real algebraic integer and D(a) has no limit points, then every algebraic conjugate of a other than a is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.
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BibTeX
@misc{campbell2026pinchs,
title = {{Pinch's conjecture on $a$-convexity}},
author = {John M. Campbell},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30771},
eprint = {2609.30771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30771},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pinch's conjecture on a-convexity
AU - John M. Campbell
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30771
UR - https://arxiv.org/abs/2609.30771
AB - For a in ℝ, a subset V contained in ℝ^n is said to be a-convex if x, y ∈ V Longrightarrow a x + (1−a) y ∈ V. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the a-convex hull of V is the intersection of all of the a-convex subsets of ℝ^n that contain V, and Pinch also defines D(a) as the a-convex hull of { 0, 1 } in ℝ^1. Pinch conjectured that if a is a totally real algebraic integer and D(a) has no limit points, then every algebraic conjugate of a other than a is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.
ER -