Question

Let f:Rn⟶R be any function, not necessarily convex. Show that f∗∗(x)≤f(x) for all x.

Answer 1

To show that f∗∗(x)≤f(x) for all x, we need to understand the concept of the convex conjugate function. The convex conjugate of a function is always convex, and therefore, for any function f:Rn⟶R, the convex conjugate f∗∗(x) will always be less than or equal to f(x) for all x.

Step-by-step explanation:

The given question involves the concept of the convex conjugate function, denoted as f∗∗(x).

To show that f∗∗(x)≤f(x) for all x, we need to understand the properties of the convex conjugate function. The convex conjugate of a function is always convex, but the original function may not be convex.

Therefore, for any function f:Rn⟶R, the convex conjugate f∗∗(x) will always be less than or equal to f(x) for all x.