Does a concave/convex real valued function with a nonempty, convex and compact domain has a maximizer?

Question

Let $X$ be a nonempty, convex and compact subset of $\mathbb{R}$ and $f:X\rightarrow \mathbb{R}$ be a real-valued function.

Does $f$ always has a maximizer in $X$ if it is concave?

Does $f$ always has a maximizer in $X$ if it is convex?

Since $X$ is a closed and bounded set, I am very tempted to say yes in the first case, but I don't have a clue how to demonstrate it.

In the second case, I tried to give a counterexample using some function that explodes like $f(x)=x^{-1}$ and $X=[0,1]$ but I'm having problems with the extreme values (if $f(0)$ assumes any real value $f$ cease to be convex).

Could anyone help me, please?

Answer 1

1. -I had a wrong example, thx L_. for pointing it out and David M. for having a correct example-

2. Yes. The maximum is at one of the endpoints of the interval $X$ (as otherwise the definition of convexity applied to the endpoints is a contradiction).

Answer 2

The answers to this are largely found in Kurdila and Zabarankin's book Convex Functional Analysis. Here, we have five theorems. Note, the point of these theorems is to state that the infimum exists and is attained, therefore the minimum exists. I highly recommend this book for establishing good conditions for when we actually have minima.

Theorem 7.3.1 (Generalized Weierstrass Theorem). Let $(X,\tau)$ be a compact topological space, and let $f: X\rightarrow{\bar{\mathbb{R}}}$ be a proper, lower semicontinuous functional. Then, there exists $x_0\in X$ such that $$ f(x_0)=\inf\limits_{x\in X} f(x) $$

Theorem 7.3.4. Let $X$ be a reflexive Banach space, and let $f: M\subseteq X\rightarrow\bar{\mathbb{R}}$ be weakly sequentially lower semicontinuous over the bounded and weakly sequentially closed subset M. Then there exists $x_0\in M$ such that $$ f(x_0) = \inf\limits_{x\in M} f(x) $$

Theorem 7.3.5. Let $X$ be a reflexive Banach space and suppose that $f: M\subseteq X\rightarrow\bar{\mathbb{R}}$ be weakly sequentially lower semicontinuous over the bounded, convex, closed subset M. Then there exists $x_0\in M$ such that $$ f(x_0) = \inf\limits_{x\in M} f(x) $$

Theorem 7.3.6. Let $X$ be a reflexive Banach space and suppose that $f: M\subseteq X\rightarrow\bar{\mathbb{R}}$ is Gateaux differentiable on the closed, convex and bounded subset M. If ${\it any}$ of the the following three conditions holds true,

1. $f$ is convex over $M$,
2. $Df$ is monotone over $M$,
3. $D^2f$ is positive over $M$,

all three conditions hold and there exists $x_0\in M$ such that $$ f(x_0) = \inf\limits_{x\in M} f(x) $$

Theorem 7.3.8. Let $X$ be a reflexive Banach space and suppose that $f: M\subseteq X\rightarrow\bar{\mathbb{R}}$ is Gateaux differentiable and coercive on the nonempty, close, convex set $M$. If one of the the following three conditions holds true,

1. $f$ is convex over $M$,
2. $Df$ is monotone over $M$,
3. $D^2f$ is positive over $M$,

all three conditions hold and there exists $x_0\in M$ such that $$ f(x_0) = \inf\limits_{x\in M} f(x) $$