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Flip-graph non-convexity for once-punctured polygons — arXiv:2609.01412

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.01412
License (as archived): CC0-1.0
Archived: 2026-10-17T07:06:29.052Z
SHA-256 of the archived text: 4c90df907015d18e09922634c3834baea6db6fdc1ad3bb42bb0a72bcd1b9cb55
Stable URL of this page: https://metraipolis.com/documento/dc_a464b507a45a

How to cite

Flip-graph non-convexity for once-punctured polygons — arXiv:2609.01412 — arXiv (en), https://arxiv.org/abs/2609.01412 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_a464b507a45a (Metraipolis archive dc_a464b507a45a, sha256 4c90df907015d18e09922634c3834baea6db6fdc1ad3bb42bb0a72bcd1b9cb55)

BibTeX

@misc{dca464b507a45a,
  title = {Flip-graph non-convexity for once-punctured polygons — arXiv:2609.01412},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.01412},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_a464b507a45a; sha256 4c90df907015d18e09922634c3834baea6db6fdc1ad3bb42bb0a72bcd1b9cb55},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.01412. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The set of the triangulations with vertex set $X$ of a simple polygon $\mathrm{P}$ can be structured into a flip-graph $\mathcal{F}(\mathrm{P},X)$ whose edges connect two triangulations that differ by a single arc. The geometry of flip-graphs has been thoroughly studied and it is known that the subgraph $\mathcal{F}_\varepsilon(\mathrm{P},X)$ induced by the triangulations that contain a given arc $\varepsilon$ is strongly convex in $\mathcal{F}(\mathrm{P},X)$ when $\mathrm{P}$ is convex and $X$ contains no puncture (points in the interior of $\mathrm{P}$) and at most one flat vertex (points in the interior of an edge). When $X$ contains at least two punctures or flat vertices, it is also known that this strong convexity property fails. Here, we close the last open case by showing that, for any convex polygon with sufficiently many vertices, one can always place a single puncture in $X$ in such a way that $\mathcal{F}_\varepsilon(\mathrm{P},X)$ is not strongly convex in $\mathcal{F}(\mathrm{P},X)$. We prove a similar result for simple polygons with a single reflex vertex. The main ingredients in our proofs are a decomposition lemma for a class of $3$-dimensional triangulations and a hyperbolic volume argument regarding their embedding into $\mathbb{H}^3$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-02 in math.CO).
Categorie arXiv: Metric Geometry (math.MG) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.01412 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T06:52:59.716Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-02?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).