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math.MG2025
On asymptotic Lebesgue's universal covering problem
Andrii Arman, Andriy Bondarenko, Andriy Prymak +1
A universal cover in is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for th…
math.MG2025
Hadwiger's conjecture for cap bodies
Andrii Arman, Jaskaran Singh Kaire, Andriy Prymak
Hadwiger's covering conjecture states that every -dimensional convex body can be covered by at most of its smaller positive homothetic translates, with copies requir…
math.MG2025
Illumination number of 3-dimensional cap bodies
Andrii Arman, Jaskaran Singh Kaire, Andriy Prymak
The illumination conjecture asserts that any convex body in -dimensional Euclidean space can be illuminated by at most external light sources or parallel beams of light. D…