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Sets homothetic to intersections of their translates
Author(s) -
McMullen P.
Publication year - 1978
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300009505
Subject(s) - mathematics , homothetic transformation , intersection (aeronautics) , combinatorics , complement (music) , kernel (algebra) , boundary (topology) , star (game theory) , regular polygon , set (abstract data type) , euclidean space , convex set , pure mathematics , discrete mathematics , mathematical analysis , geometry , biochemistry , chemistry , programming language , convex optimization , complementation , computer science , engineering , gene , phenotype , aerospace engineering
Let S be a compact set in some euclidean space, such that every homo‐thetic copy λS of S , with 0 < λ < 1, can be expressed as the intersection of some family of translates of S . It is shown that S has this property precisely when it is star‐shaped, and is such that every point in the complement of S is visible from some point (necessarily on the boundary) of the kernel of S . Alternatively, S can be characterized as a compact star‐shaped set, whose maximal convex subsets are cap‐bodies of its kernel.