# How to prove the inverse image of affine convex function is still a convex?

Consider a convex set $$S ⊆ ℝ^n$$ and an affine function $$f(x)=Ax+b$$. Then if the image of $$S$$ under $$f$$,$$f(S)$$={$$f(x)|x \in S$$} , is convex. Then the inverse image of the convex set C,$$f^{-1}(C)=\{x|f(x) \in C$$}, is convex set.
Hint:Let $$y_1$$ and $$y_2 \in C.$$Then there exist $$x_1$$ and $$x_2 \in f^{-1}(C)$$.Show that $$f^{-1}(C)$$ is convex.

How to prove the inverse of affine convex function is still a function?Because for me,i intuitively think the inverse image of the convex is still a convex.

My proof is as below,i am not sure whether it is right or not

For two elements of the preimage $$x_1,x_2\in f^{-1}(C)$$ and $$t\in[0,1]$$, you have to show that $$tx_1+(1-t)x_2\in f^{-1}(C)$$, i.e. that there exists an element $$y\in C$$, such that $$tx_1+(1-t)x_2=f(y)$$.

Since $$x_1,x_2\in f^{-1}(C)$$ there exist $$y_1,y_2\in C$$, such that $$f(y_1)=x_1,f(y_2)=x_2$$. Now use the definition of $$f$$ to find $$y$$ and you are done.

So \begin{align} f(tx_1+(1-t)x_2) &=A(tx_1+(1-t)x_2)+b\\ & = t(Ax_1+b)+(1-t)(Ax_2+b)\\ & = tf(x_1)+(1-t)f(x_2)\\ & = f(y) \end{align} And the question said $$f(x)$$ is a convex,that is,$$f(tx_1+(1-t)x_2)$$ is convex,so $$f(y)$$ is convex,that is $$f^{-1}(C)$$ is convex

Is this proof right?

• If you wish to show curly braces $\{ \ldots \}$ when using MathJax, you can use \{ and \}. – Theo Bendit Oct 25 '18 at 7:26
• You don't need $f$ to have an inverse. The inverse image $f^{-1}(S)$ is defined, regardless of whether $f$ is invertible, to be the set of points $x$ such that $f(x) \in S$. For example, if $f : \mathbb{R} \to \mathbb{R}$ is constantly $y$, then $f^{-1}(S)$ will either be $\emptyset$ if $y \notin S$ or $\mathbb{R}$ if $y \in S$. Note that $f^{-1}$ is not a function, as $f$ is very far from being one-to-one! – Theo Bendit Oct 25 '18 at 7:29
• @TheoBendit so the proof if this doesn't have relation with $f(x)=Ax+b$?i just said that there exist s $x_1$ and $x_2 \in C$,and C is convex set,so the $f^{-1}(C)$ must be convex too? – electronic component Oct 25 '18 at 7:57
• It does relate to $f$, but it doesn't require there to be a function $f^{-1}$. I may have misunderstood your issue with this question, but you seemed to be interested in showing $f^{-1}$ is still a function (which, in general, it won't be). – Theo Bendit Oct 25 '18 at 8:00
• not quite. My hint was a bit misleading and there was some mixup with $y$ and $x$, i finished it up in my answer. – weee Oct 26 '18 at 20:42

For two elements of the preimage $$x_1,x_2\in f^{-1}(C)$$ and $$t\in[0,1]$$, you have to show that $$tx_1+(1-t)x_2\in f^{-1}(C)$$, i.e. that $$f(tx_1+(1-t)x_2)\in C$$. ($$S$$ is convex so $$tx_1+(1-t)x_2\in S$$ and we can apply $$f$$)

Since $$x_1,x_2\in f^{-1}(C)$$ we have $$f(x_1)\in C,f(x_2)\in C$$.

Edit: to finish up, we calculate \begin{align} f(tx_1+(1-t)x_2)&=A(tx_1+(1-t)x_2)+b\\ &=t(Ax_1+b)+(1-t)(Ax_2+b)\\ &=tf(x_1)+(1-t)f(x_2). \end{align} Since $$C$$ is convex, and $$f(x_1)\in C,f(x_2)\in C$$ we have also $$tf(x_1)+(1-t)f(x_2)\in C$$ and are done.

• the definition of $f$ is $Ax+b$,but how can we find $y$ by using this function?how do you know the $f(y_1)=x_1$ and $f(y_2)=x_2$ from the $tx_1+(1-t)x_2=f(y)$? – electronic component Oct 26 '18 at 2:02