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We define a class of L-convex-concave subsets of RP n , where L is a projective subspace of dimension l in RP n . These are sets whose sections by any (l+1)-dimensional space L containing L are convex and concavely depend on L . We introduce an Lduality for these sets, and prove that the L-dual to an L-convex-concave set is an L * -convex-concave subset of (RP n ) * . We discuss a version of Arnold hypothesis for these sets and prove that it is true (or wrong) for an L-convex-concave set anddoi:10.17323/1609-4514-2003-3-3-1013-1037 fatcat:p7uz6labevajpgca6wssfxu2ka