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A conjecture by Dol\u2019nikov is that, under these conditions, there is always some \n \n $$j \\in \\{ 1,2,3 \\}$$<\/jats:tex-math>\n \n \n j<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 3<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that \n \n $$\\mathcal {F}_{j}$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can be pierced by 3 points. In this paper we prove a stronger version of this conjecture when K<\/jats:italic> is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conjecture is true with 8 piercing points instead of 3. Along the way we prove more general statements both in the plane and in higher dimensions. A related result was given by Mart\u00ednez-Sandoval, Rold\u00e1n-Pensado and Rubin. They showed that if \n \n $$\\mathcal {F}_{1}, \\dots , \\mathcal {F}_{d}$$<\/jats:tex-math>\n \n \n \n F<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n F<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are finite families of convex sets in \n \n $$\\mathbb {R}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that for every choice of sets \n \n $$C_{1} \\in \\mathcal {F}_{1}, \\dots , C_{d} \\in \\mathcal {F}_{d}$$<\/jats:tex-math>\n \n \n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \u2208<\/mml:mo>\n \n F<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n C<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n \u2208<\/mml:mo>\n \n F<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> the intersection \n \n $$\\bigcap _{i=1}^{d} {C_{i}}$$<\/jats:tex-math>\n \n \n \n \u22c2<\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msubsup>\n \n C<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is non-empty, then either there exists \n \n $$j \\in \\{ 1,2, \\dots , n \\}$$<\/jats:tex-math>\n \n \n j<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n n<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that \n \n $$\\mathcal {F}_j$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can be pierced by few points or \n \n $$\\bigcup _{i=1}^{n} \\mathcal {F}_{i}$$<\/jats:tex-math>\n \n \n \n \u22c3<\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msubsup>\n \n F<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when \n \n $$d=2$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and also consider the problem restricted to special families of convex sets.<\/jats:p>","DOI":"10.1007\/s00454-024-00669-3","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T15:05:47Z","timestamp":1719500747000},"page":"1079-1096","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Transversals to Colorful Intersecting Convex Sets"],"prefix":"10.1007","volume":"73","author":[{"given":"Cuauhtemoc","family":"Gomez-Navarro","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2164-066X","authenticated-orcid":false,"given":"Edgardo","family":"Rold\u00e1n-Pensado","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"issue":"2","key":"669_CR1","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1007\/s00454-009-9180-4","volume":"42","author":"JL Arocha","year":"2009","unstructured":"Arocha, J.L., B\u00e1r\u00e1ny, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. 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