Concavity of f(c) in Linear Programming

[SantyClause]

Homework Statement

Let c in R^n be a parameter and consider the following function of c:

f(c)= minimize cx
subject to Ax=b
x>/= 0

where x is an n-dimensional decision vector, A is an mxn matrix, and b is an m-dimensional constant. The function f(c) is determined by solving the above linear program for a particular value of the parameter c. Show that f(c) is concave.

The Attempt at a Solution

Let x^1 be solution to the linear program when c=c^1 and x^2 be the solution when c=c^2

I need to show the following: f(Lc^1 +(1-L)c^2) >/= Lf(c^1) + (1-L)f(c^2)

the RHS is Lx^1+(1-L)x^2, but I'm not sure what to do with the LHS. It means solving an LP with an objective function of Lc^1x + (1-L)c^2x

 

[lippyka]

, but I'm not sure how to go about proving it is greater than/equal to the RHS. Any help would be appreciated.