3 problems

Chamberland's conjecture. The mapping $f$ is injective.

Alexandrov's conjecture. The image $f(\mathbb{R}^n)$ is a convex domain in $\mathbb{R}^n$, and $f$ is injective.

Let $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map. Suppose there exists an $\varepsilon>0$ such that $|\lambda|\geqslant\varepsilon$ for every eigenvalue $\lambda$ of…