Is the set of all concave functions a convex set?

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Tirioliwo · Accepted Answer

Let  S  =  {  g  :            R        n    →      R       s.t     g     concave  }For   a  ,  b  ∈  [  0  ,  1  ] and   x  ,  y  ∈            R        n   and   g  ,  f  ∈  S we have:                    (        a        f        +        (        1        −        a        )        g        )        (        b        x        +        (        1        −        b        )        y        )                            =        a        f        (        b        x        +        (        1        −        b        )        y        )        +        (        1        −        a        )        g        (        b        x        +        (        1        −        b        )        y        )                                          ≥        a        (        b        f        (        x        )        +        (        1        −        b        )        f        (        y        )        )        +        (        1        −        a        )        (        b        g        (        x        )        +        (        1        −        b        )        g        (        y        )        )                                          =        b        (        a        f        (        x        )        +        (        1        −        a        )        g        (        x        )        )        +        (        1        −        b        )        (        a        f        (        y        )        +        (        1        −        a        )        g        (        y        )        )                                          =        b        (        a        f        +        (        1        −        a        )        g        )        (        x        )        +        (        1        −        b        )        (        a        f        +        (        1        −        a        )        g        )        )        (        y        )        .            Then,   a  f  +  (  1  −  a  )  g is concave so it belongs   S and hence   S is convex set.

Is the set of all concave functions a convex set?

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