Geodesic
A criterion for checking if a convex set belongs to the union of a finite number of convex sets
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1571-1578

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A necessary and sufficient condition for the inclusion of a convex compact set in the union of a finite number of convex sets is proved. This condition obtained using the convex analysis techniques is a condition on the maximin of a given function. Using the dynamic programming, checking this condition is reduced to evaluating a set of functions and checking a condition for their values. The reduced form of the criterion is more convenient from the computational point of view. 
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     author = {D. G. Pivovarchuk},
     title = {A~criterion for checking if a~convex set belongs to the union of a~finite number of convex sets},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1571--1578},
     publisher = {mathdoc},
     volume = {49},
     number = {9},
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     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a3/}
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D. G. Pivovarchuk. A criterion for checking if a convex set belongs to the union of a finite number of convex sets. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1571-1578. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a3/