caa a22 4500 469075171 CHVBK 20180323132933.0 cr unu---uuuuu 170328e19920101xx s 000 0 eng 10.1007/BF02187827 doi (NATIONALLICENCE)springer-10.1007/BF02187827 Kupitz Yaakov Institute of Mathematics, The Hebrew University of Jerusalem, Jersusalem, Israel aut On a generalization of the Gallai-Sylvester theorem [Elektronische Daten] [Yaakov Kupitz] It is shown that ifS ⊂ ℝ d , affS=aff ℝ d , and every hyperplane spanned by (a subset of)S misses fewer thank points ofS(k≥2), then (a) #S≤km ifd=2m−1 is odd and (b) #S≤km+1 ifd=2m is even. We also fully describe the extreme sets for which equality holds in (a) or in (b). For oddd the proofs are later modified to purely algebraic ones, and carry over to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr% hAYaqeguuDJXwAKbstHrhAGq1DVbacfeGae8xHWB0aaWbaaSqabeaa% caWGKbaaaaaa!44DB! $$\mathbb{F}^d $$ , where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr% hAYaqeguuDJXwAKbstHrhAGq1DVbacfeGae8xHWBeaaa!43C5! $$\mathbb{F}$$ is an arbitrary field. For evend, (b) is generally not true when % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr% hAYaqeguuDJXwAKbstHrhAGq1DVbacfeGae8xHWBKaeyiyIKRae8xh% Hifaaa!46A1! $$\mathbb{F} \ne \mathbb{R}$$ , but we prove some weaker inequalities that do hold over arbitrary fields. Springer-Verlag New York Inc., 1992 Discrete & Computational Geometry Springer New York 7/1(1992-01-01), 87-103 0179-5376 7:1<87 1992 7 454 https://doi.org/10.1007/BF02187827 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02187827 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Kupitz Yaakov Institute of Mathematics, The Hebrew University of Jerusalem, Jersusalem, Israel aut NATIONALLICENCE 773 0- Discrete & Computational Geometry Springer New York 7/1(1992-01-01), 87-103 0179-5376 7:1<87 1992 7 454 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer