Duality and Upper Semicontinuity of Set Valued Functions

  • Lamberto Cesari
Part of the Applications of Mathematics book series (SMAP, volume 17)


We shall consider here a real valued function F(u) defined and everywhere finite on a set U of R n . If U is convex (Section 8.4), then F is said to be convex in U if u1, u2U, 0 ≤ α ≤ 1, implies F(αu1 + (1 − α)u2) ≤ αF(u1) + (1 − α)F(u2). The function F is said to be extended (Section 8.5) if we take F = + ∞ in R n U. With obvious conventions the convexity of F in R n is equivalent to the statement that U is convex and F is convex in U. As mentioned in Section 8.5, the set \( \tilde{Q} = {\text{ }}\left[ {({{z}^{0}},u) + \infty > {{z}^{0}} > F\left( u \right),u \in U} \right] \) is said to be the epigraph of F, or epi F.


Convex Function Convex Subset Half Space Convex Combination Lower Semicontinuous 
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Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  • Lamberto Cesari
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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