Let $f\colon X \to [ -\infty, +\infty]$ be proper and strictly convex. Show that $f$ has at most one minimizer. Give examples where $f$ is strictly convex, and either (i) $f$ has one minimizer; or (ii) $f$ has no minimizer.

For the strict convexity part, you may use the fact that f is strictly convex whenever its Hessian is positive definite.

I am not sure how to show that f has at most one minimizer, could someone help get me started with the question?


Recall the definition of strict convexity: $$f(\lambda x + (1-\lambda)y) < \lambda f(x) + (1-\lambda) f(y) \quad \forall x,y\in\mathop{\textrm{dom}}(f), ~ x\neq y, ~\lambda\in(0,1)$$

The strictly convex functions $f(x)=e^x$ and $f(x)=x^2$ have exactly zero and one minimizers, respectively.

Now suppose we claim that a strictly convex function $f$ has two or more minimizers. Let $x$ and $y$ be two of those minimizers ($x\neq y$); obviously, $f(x)=f(y)$. Then along the secant between these two points, we have $$f(\lambda x + (1-\lambda)y) < \lambda f(x) + (1-\lambda) f(y) = \lambda f(x) + (1-\lambda) f(x) = f(x).$$ Hence we have $f(z)<f(x)$, where $z$ is any point strictly between $x$ and $y$---a contradiction. It is not possible therefore for a strictly convex function to have two minimizers (or more).

EDIT: I note that you're using an extended real convention. For the most part, the above argument is unchanged. However, we do need to consider convex functions that obtain $f(x)=-\infty$ at some $x$. Convince yourself of this: that the only convex functions that obtain the value $-\infty$ do so on their entire domain; and the only strictly convex functions that do so have just a single point as their domain.


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