For some strictly convex functions f 1 ( x 1 ) , f 2 ( x 2 ) , . . . , f n ( x n ), it seems intuitive that for a given sum of x 1 + x 2 + . . . + x n = X, where x 1 , x 2 ,..., x n ≥ 0 ∃ k ∈ { 1 , 2 , … , n } , f k ( X ) = max { ∑ i = 1 n f i ( x i ) : ∑ i = 1 n x i = X and ∀ i , x i ≥ 0 } Since these functions are convex, the sum of these functions is greatest when the sum of arguments is equal to the argument of just one function. This seems intuitive, but I can't think of how to argue this in formal maths. Any suggestions?

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For some strictly convex functions       f    1    (      x    1    )  ,      f    2    (      x    2    )  ,  .  .  .  ,      f    n    (      x    n    ), it seems intuitive that for a given sum of       x    1    +      x    2    +  .  .  .  +      x    n    =  X, where       x    1  ,       x    2  ,...,       x    n    ≥  0  ∃  k  ∈  {  1  ,  2  ,  …  ,  n  }  ,        f    k    (  X  )  =  max      {          ∑              i        =        1            n                      f        i            (              x        i            )            :          ∑              i        =        1            n              x      i        =    X     and     ∀    i    ,              x      i        ≥    0        }  Since these functions are convex, the sum of these functions is greatest when the sum of arguments is equal to the argument of just one function. This seems intuitive, but I can&#039;t think of how to argue this in formal maths. Any suggestions?

exorteygrdh · Accepted Answer

You can use the following theorem: if the maximum of a convex function on a convex set is attained, then it is attained at an extreme point.In your case, the convex set is a simplex, which is a compact set. The function   F  (  x  )  =      ∑          i      =      1        n        f    i    (      x    i    ) is convex, and thus continuous. Therefore, the maximum is attained by Weirstrass. The constraints define a convex polytope, whose extreme points are its vertices, which are   (  X  ,  0  ,  …  ,  0  )  ,     (  0  ,  X  ,  0  ,  …  ,  0  )  ,  …  ,  (  0  ,  …  ,  0  ,  X  ). The result you wish follows from the fact that the maximum is attained at one of those vertices.

For some strictly convex functions f 1 </msub> ( x 1 </msub>

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