Convex bodies of constant width with exponential illumination number

2023/04/20 by Arman, Andrii, Bondarenko, Andriy, Prymak, Andriy
#52A40 #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 52C17 #Secondary 52A20

paper · doi:10.48550/arxiv.2304.10418

Abstract

We show that there exist convex bodies of constant width in 𝔼ⁿ with illumination number at least (cos(π/14)+o(1))^-n, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in 𝔼ⁿ which cannot be covered by (2/√(3)+o(1))ⁿ balls of diameter 1, improving a result by J. Bourgain and J. Lindenstrauss.

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