On the illumination of centrally symmetric cap bodies in small dimensions

Ivanov, Ilya; Strachan, Cameron

doi:10.1007/s00022-020-00568-x

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2022

2024

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Cited by 3 publications

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“…(This is not a new result. It was proved via a dual method in [16].) On the other hand, any 4dimensional centrally symmetric cap body can be illuminated by 12…”

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confidence: 98%

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Illuminating spiky balls and cap bodies

Bezdek¹,

Ivanov²,

Strachan³

2022

Preprint

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The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the illumination numbers of 2-illuminable spiky balls as well as centrally symmetric cap bodies.In particular, we prove the Illumination Conjecture for centrally symmetric cap bodies in sufficiently large dimensions by showing that any d-dimensional centrally symmetric cap body can be illuminated by < 2 d directions in Euclidean d-space for all d ≥ 20. Furthermore, we strengthen the latter result for 1-unconditionally symmetric cap bodies.

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“…(This is not a new result. It was proved via a dual method in [16].) On the other hand, any 4dimensional centrally symmetric cap body can be illuminated by 12…”

mentioning

confidence: 98%

“…We close this section with a strengthening of Corollary 7 for 1-unconditionally symmetric cap bodies. Recall that according to [16] if K is a 1-unconditionally symmetric cap body in E 4 , then I(K) ≤ 8. Theorem 8.…”

mentioning

confidence: 99%