$f:U\rightarrow V$ a $C^2$ diffeo, $\forall\, a\in U, \exists r>0$ such that $f\left(B_{a}(\epsilon)\right)$ is convex $\forall\epsilon\leq r$

Question

Let $U,V$ open sets in $\Bbb{R}^n$, $f:U\rightarrow V$ a diffeomorphism of class $C^2$. I need to prove that, for all $a\in U$, exists $r>0$ such that the image of the open ball centered at $a$ with radius $\epsilon$ is convex, for all $\epsilon\leq r$.

My idea is very simple, and that is why I think that I'm forgetting something. I found some different answers for that question, like this one. But I want to know what is wrong about my thinking.

Well, in fact I only used the fact that $f^{-1}$ is continuous and $f$ is surjective. If $a\in U$, let $A$ an open ball of $a$. So, $f(A)$ is a open set. So, there exists $r>0$ such that $B_{f(a)}(r)\subset f(A)$. So, I can use these balls as the convex questions.

What is wrong? And what can I do to solve the question? I did not understand the answerd linked above, since it uses some statements about the Hessian, and I didn't study hessians yet. The context is inverse funcion theorem.

I don't understand "use these balls as the convex questions". Nor do I see what your proof has to do with the question. You're asked to prove that the images of sufficiently small balls are convex. You prove that the images of arbitrary open sets have balls as subsets. What's the connection? — Andreas Blass, Apr 08 '20 at 02:00
@MateusRocha você está cursando a disciplina MA720, correto? — rfloc, Apr 08 '20 at 03:03
@rfloc estou. A solução que achei foi usar um teorema que diz que uma função $C^2$ é convexa em um conjunto convexo $\iff$ sua hessiana é semi positiva definida no interior deste conjunto convexo — Mateus Rocha, Apr 09 '20 at 17:37

$f:U\rightarrow V$ a $C^2$ diffeo, $\forall\, a\in U, \exists r>0$ such that $f\left(B_{a}(\epsilon)\right)$ is convex $\forall\epsilon\leq r$

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