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The affine Tverberg theorem revisited
Abstract
We prove the Bárány--Kalai--Tverberg conjecture extending the affine version of Tverberg's theorem to simplicial balls and polytopes other than the simplex: For any affine map ϕ: P→ ℝd from a convex polytope P of dimension N=(d+1)(r−1) with r≥ 2, there exist r pairwise disjoint faces F1,…,Fr⊂∂ P such that ϕ(F1)∩…∩ϕ(Fr)≠∅. A similar statement holds for a simplicial ball P with ϕ assumed affine on all of its faces.
Related
BibTeX
@misc{avvakumov2026affine,
title = {{The affine Tverberg theorem revisited}},
author = {Sergey Avvakumov and Roman Karasev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12483},
eprint = {2609.12483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12483},
}
RIS
TY - UNPB
TI - The affine Tverberg theorem revisited
AU - Sergey Avvakumov
AU - Roman Karasev
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12483
UR - https://arxiv.org/abs/2609.12483
AB - We prove the Bárány--Kalai--Tverberg conjecture extending the affine version of Tverberg's theorem to simplicial balls and polytopes other than the simplex: For any affine map ϕ: P→ ℝ^d from a convex polytope P of dimension N=(d+1)(r−1) with r≥ 2, there exist r pairwise disjoint faces F_1,…,F_r⊂∂ P such that ϕ(F_1)∩…∩ϕ(F_r)≠∅. A similar statement holds for a simplicial ball P with ϕ assumed affine on all of its faces.
ER -