On a Gallai‐type problem and illumination of spiky balls and cap bodies
Abstract We show that any finite family of pairwise intersecting balls in can be pierced by points improving the previously known estimate of . As a corollary, this implies that any 2‐illuminable spiky ball in can be illuminated by directions. For the illumination number of convex spiky balls, that is, cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of , which can be compared with the previous established only for the centrally symmetric cap bodies. We also prove the lower bounds of for the three problems above.
Authors
- Danylo Radchenko (ORCID: https://orcid.org/0000-0003-1457-6214)
- Andrii Arman (ORCID: https://orcid.org/0000-0003-1462-5733)
- Andriy Prymak (ORCID: https://orcid.org/0000-0002-5022-0154)
- Andriy Bondarenko (ORCID: https://orcid.org/0000-0002-2303-8525)
Institutions
- Norwegian University of Science and Technology (NO)
- Université de Lille (FR)
- Laboratoire Paul Painlevé (FR)
- University of Manitoba (CA)
Publication Details
- Journal
- Mathematika
- Published
- 2025-03-25
- DOI
- https://doi.org/10.1112/mtk.70017
- Citations
- 1
- Primary Topic
- Point processes and geometric inequalities
- Type
- article
- Field-Weighted Citation Impact
- 3.73
Funders
- Pacific Institute for the Mathematical Sciences
- Norges Forskningsråd
- Natural Sciences and Engineering Research Council of Canada
- European Research Council
- Division of Mathematical Sciences