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DTSTART;TZID=America/Los_Angeles:20250211T121500
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DTSTAMP:20260516T061120
CREATED:20250206T203702Z
LAST-MODIFIED:20250206T203729Z
UID:3689-1739276100-1739279400@colleges.claremont.edu
SUMMARY:On the illumination problem for convex sets (Lenny Fukshansky\, CMC)
DESCRIPTION:Let K be a compact convex set in the Euclidean space R^n. How many lights are needed to illuminate its boundary? A classical conjecture of Boltyanskii (1960) asserts that 2^n lights are sufficient to illuminate any such set K. While this is still open\, an earlier observation of Hadwiger (1945) guarantees that if K has smooth boundary\, then n+1 lights are sufficient: we only need to position these lights at the vertices of a simplex containing K in its interior. In fact\, this observation allows us to estimate how far from K these lights need to be. A more delicate problem arises if we insist on placing the lights at points of a fixed lattice L: how far from K must the lights be then? We discuss this problem\, producing a bound on this distance\, which depends on certain orthogonality and symmetry properties of the lattice in question. Interestingly\, for some nice classes of lattices\, a bound independent of L can be produced.
URL:https://colleges.claremont.edu/ccms/event/on-the-illumination-problem-for-convex-sets/
LOCATION:Estella 2113
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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