Geodesic
The illumination conjecture for spindle convex bodies
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 181-187

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A subset of the $d$-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent $d$-dimensional closed balls. A spindle convex body is called a “fat” one if it contains the centers of its generating balls. The main result of this paper is a proof of the illumination conjecture for “fat” spindle convex bodies in dimensions greater than or equal to 15. 
@article{TM_2011_275_a10,
     author = {K\'aroly Bezdek},
     title = {The illumination conjecture for spindle convex bodies},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {181--187},
     publisher = {mathdoc},
     volume = {275},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_275_a10/}
}
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Károly Bezdek. The illumination conjecture for spindle convex bodies. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 181-187. http://geodesic.mathdoc.fr/item/TM_2011_275_a10/