**EDIT:** Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f\_1,\dots, f\_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there exist directions $\xi\_1^\pm,\dots,\xi\_n^\pm\in \Sigma\_p$ such that $f'\_i(\xi\_i^+)>1,\, f'\_i(\xi^-\_i)<-1$ and $|f'\_i(\xi\_j^\pm)|<\frac{1}{100 n}$ for $i\ne j$.


**Is it true that the map
$$(f\_1,\dots,f\_n)\colon X\to \mathbb{R}^n$$
is injective in a small neighborhood of $p$?**


Remark. This question is motivated by an attempt to understand the paper "Elements of Morse theory on Alexandrov spaces" by Perelman (1993). He apparently does not explain the initial step in his induction: given an admissible map $X^n\to \mathbb{R}^n$, then it is a local homeomorphism in a neighborhood of any regular point. Thus if one has a proof of injectivity in this statement, it will be great.



A property of concave functions on Alexandrov spaces
The base of Perelman's induction for Main Theorem 1.4 (is this what you're talking about?) is in fact a collection of trivial statements about admissible maps $X^n \to \mathbb{R}^{n+1}$.


To move to the next step, which you're looking for, requires an application of Property 1.3(c). The complementing function $g\_{n+1}$ is constantly zero.


If you want to understand issues around the Stability Theorem, I strongly recommend Kapovitch's [expository article](http://www.math.toronto.edu/vtk/stability-final3.pdf). In particular, check out Corollary 6.10. The map is actually bi-Lipschitz.


