smooth convex function

Question

Suppose $f(x, y)$ is a smooth convex function and $f(0, 0) = f(1, 0) = f(0, 1) = 0$.

(a) What do you know about $f (1/2, 1/2)$ ?

(b) What do you know about the second derivatives $a=∂^2 f/∂x^2 $ , $b=∂^2 f/∂x ∂y $ and $c=∂^2 f/∂y^2 $ ?

Answer 1

You should familiarize yourself with the various definitions of convexity of a function (which is related to, but not the same as the definition of convexity of a set).

For a convex function of two variables, the definition of convexity is $$f(\lambda x + (1-\lambda) x', \lambda y + (1-\lambda) y') \le \lambda f(x,y) + (1-\lambda) f(x', y'), \qquad \text{for all $(x,y), (x',y')$ and all $\lambda \in [0,1]$}$$ Graphically, this means that the secant line connecting $(x,y, f(x,y))$ and $(x', y', f(x',y'))$ lies above the graph of $f$. Applying this definition to part (a) will immediately tell you something about $f(1/2, 1/2)$.

If $f$ is smooth and convex, then the Hessian is positive semi-definite. [You should do some investigation on why this follows from the definition of convexity.] Knowing this, solving (b) just involves writing the condition "the Hessian of $f$ is positive semi-definite" in terms of $a, b, c$.