Proof Verification: Showing a function is affine if its convex and concave

Question

I know this question has been asked several times but the answers don't really make sense to me (I'll explain misunderstandings:)

Question: Suppose that a function $f: \mathbb R^n \rightarrow \mathbb{R}$ is both concave and convex. Prove that $f$ is an affine function. My solution uses: How to prove convex+concave=affine? as inspiration. However, I'm not sure if I did it correctly especially for the negative cases and I'm not exactly sure if I have showed that $g$ is linear in all cases.

et $g(x)=f(x)-a$, where $a=f(0)$. Thus $g(0)=0$. Also since $f$ is convex and concave, $g$ is convex and concave as well. Thus for $x,y \in \mathbb R^n $, $0 \leq \lambda \leq 1$, by inequalities resulting from convexity and concavity we have: $g(\lambda x +(1-\lambda) y)$=$\lambda g(x) +(1-\lambda)g(y)$. Does this mean $g$ is linear (why?, I don't get this from looking at the other stack-exchange posts) and hence $f$ is affine.

Case 2: $\lambda >1$. Note $x = (1/\lambda ) (\lambda x) + (1 - 1/\lambda) (0)$. Note then $1/ \lambda \in [0,1]$. Thus $g(x)=g((1/\lambda) (\lambda x) + (1 - 1/\lambda) (0))$=$1/\lambda \cdot g(\lambda x)+$$(1- 1/\lambda) \cdot g(0) $. This means $g(x)=1/\lambda * g(\lambda x)$. Hence $g(\lambda x)=\lambda g(x)$.

Case 3: $\lambda <0$. Not sure what to do now....

Perhaps I could do :

Case : $\lambda \leq -1$.

$$x=(-1/\lambda)(-\lambda x)+(1+1/\lambda)(0)$$ Note that $-1/\lambda \in [0,1]$

$g(x)=g((-1/\lambda)(-\lambda x) + (1 + 1/\lambda) (0))$=$-1/\lambda \cdot g(-\lambda x)+$$(1+1/\lambda) \cdot g(0) $. This means $g(x)=-1/\lambda * g(-\lambda x)$. Hence $g(-\lambda x)=-\lambda g(x)$.

Case: $-1< \lambda <0$ Note $-\lambda \in [0,1]$. Thus $x=(-\lambda)(-1/\lambda \cdot x)+(1+\lambda)(0)$

$g(x)=g((-\lambda) (-1/\lambda \cdot x) + (1 + \lambda) (0))$=$-\lambda \cdot g(-1/\lambda \cdot x)+$$(1+\lambda) \cdot g(0) $. This means $g(x)=-\lambda * g(-1/\lambda \cdot x)$. Hence $g(-1/\lambda \cdot x)=-1/ \lambda \cdot g(x)$.

Any help would much appreciated. Thanks.

Thus, linear in all cases...not exactly sure if this is correct at all.

Answer 1

Your proof is (mostly) finished after your case $0\leq \lambda \leq 1$! In other words, the other cases you consider for $\lambda$ are not required.

I will go into more detail as to why this is the case and hopefully alleviate some confusions. First of all I will mention the very important definition:

A function $g:\mathbb{R}^n\rightarrow \mathbb{R}$ is linear if for all scalars $\gamma \in \mathbb{R}$ and $x\in\mathbb{R}^n$ we have $g(\gamma x) = \gamma g(x)$ and for all $x,y \in \mathbb{R}^n$ we have $g(x+y) = g(x) + g(y)$.

So in order to prove a function is linear we need to show that both these conditions hold. Returning to your problem:

You have shown that if a function $g$ (such that $g(0) = 0$) is both convex and concave then for every $\lambda\in[0,1]$ and for every $x,y\in \mathbb{R}^n$ we have \begin{equation} g(\lambda x + (1-\lambda)y) = \lambda g(x) + (1-\lambda)g(y). \end{equation} I claim that this enough to show that $g$ is linear. Note that in the above equation this holds for any choice of $\lambda \in[0,1]$ (and $x,y\in\mathbb{R}^n$) and so we are free to pick $\lambda$ (and $x$ and $y$) as we please! We will first show that $g(\gamma x) = \gamma g(x)$ for every $\gamma \in \mathbb{R}$ and $x\in\mathbb{R}^n$:

Note that if $\gamma = 0$ or $\gamma = 1$ then our claim is trivially true (since $g(1\cdot x) = 1\cdot g(x)$ and $0 = g(0\cdot x) = 0\cdot g(x)$ by definition of $g$). Consider $\gamma\in(0,1)$ then, using our above equation (which we are allowed to do since $\gamma\in(0,1)$ with $\lambda = \gamma$ and $y=0$, \begin{equation} g(\gamma x) = g(\gamma x + (1-\gamma)\cdot0) = \gamma g(x). \end{equation} If $\gamma > 1$ then $0<\frac{1}{\gamma} < 1$ and we can use our above equation again. Using the equation with $\lambda = 1/\gamma$, $x = \gamma X$ and $y=0$ we obtain \begin{equation} \gamma g(X) = \gamma g\left(\frac{1}{\gamma} (\gamma X) + (1-\frac{1}{\gamma})\cdot0\right) = \gamma\cdot 1/\gamma\cdot g(\gamma X) = g(\gamma X), \end{equation} by our previous proof. Thus we have shown that $g(\gamma x) = \gamma g(x)$ for $\gamma\geq 0.$ To show that this holds for $\gamma < 0$ too, we will use the equation above again with $\lambda = 1/2$, $x = 2X$ and $y = -2X$. Then \begin{equation} 0 = g(0) = g\left(\frac{1}{2}\cdot 2X - \frac{1}{2}\cdot2X)\right) = \frac{1}{2}g(2X) + \frac{1}{2}g(-2X) = g(X) + g(-X), \end{equation} where in the last step we used our previously proven result that $g(\gamma x) = \gamma g(x)$ for $\gamma \geq 0$. This implies that $g(-x) = -g(x).$ Thus for $\gamma < 0$ we have (since $-\gamma > 0$), $-\gamma g(x) = g(-\gamma x) = -g(\gamma x)$ by our previous arguments. And so $\gamma g(x) = g(\gamma x)$.

We now only need to prove that $g(x + y) = g(x) +g(y)$ for every $x,y\in\mathbb{R}^n$. Similar tricks to before can be used, but I will leave the details for you. (Hint: Use $\lambda =1/2$ and a tactical choice of $x$ and $y$).

We are now finished.