Let $U,V\subset \mathbb R^m$ be open subsets and $f:U\to V$ a diffeomorphism of class $C^k$. I want to prove that for every point $a\in U$, there is $r\gt 0$ such that for every ball $B=B(a;\delta)$ with $\delta\le r$ the image $f(B)$ is a convex set. I don't know even how to begin to tackle this question, I need a hint or some suggestions.
How to prove this set is convex
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real-analysis