Let f : R n × R m → R be a continuous function. Suppose that there exists an a 0 ∈ R m such that f ( x , a 0 ) is strictly concave in x. Is it true that there exists a ball A around a 0 such that f ( x , a ) is concave in f ( x , a ) for a ∈ A?

Question

Let   f  :            R        n    ×            R        m    →      R   be a continuous function. Suppose that there exists an       a    0    ∈            R        m   such that   f      (    x    ,          a      0        )   is strictly concave in   x. Is it true that there exists a ball   A around       a    0   such that   f      (    x    ,    a    )   is concave in   f      (    x    ,    a    )   for   a  ∈  A?

hamputlnf · Accepted Answer

This not true; essentially all we need to do is set  f  (  x  ,  a  )  =  −      x    2    +  a  g  (  x  )and pick   g  (  x  ) to be any function whose second derivative is not bounded above. Then      d          d      x            d          d      x        f  (  x  ,  a  )  =  −  2  +  a      g    ″    (  x  )  .Since       g    ″    (  x  ) is not bounded above, there is some   x for which this is positive for any   a  &amp;gt;  0.An explicit example of this would be  f  (  x  ,  a  )  =  −      x    2    +  a  sin  ⁡  (      x    2    )  .We can even do a little worse - consider a function like   f  (  x  ,  a  )  =  −      x    2    +  a      |    x      |  . This gives a counterexample even you wanted   f to be a function   f  :  [  −  1  ,  1  ]  ×      R    →      R  .

Let f:R^n x R^m->R be a continuous function. Suppose that there exists an a_0 in R^m such that f(x,a_0) is strictly concave in x. Is it true that there exists a ball A around a_0 such that f(x,a) is concave in x for a in A?

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