Projections from Tubular Neighborhoods

[Goklayeh]

Hello! Could anybody give me some hint with the following problem? Consider a smooth, compact embedded submanifold M = M^m\subset \mathbb{R}^n, and consider a tubular neighborhood U = E(V)\supset M, where E: (x, v) \in NM \mapsto x + v \in M is a diffeomorphism from a open subset of the normal bundle NM of the form V = \left\{(x, v) \in NM \: : \: \left| v \right| < \delta\right\}. We know that r:=\pi \circ E^{-1} is a smooth retraction, where \pi:(x, v) \in NM \mapsto x \in M is the projection. How can I prove that if y \in U, where U is a sufficiently small tubular neighborhood, then r(y) realizes the minimum of the distance from the points of M? I just proved, following Lee's hints, that if y \in \mathbb{R}^n has a closest point x \in M, then y - x \in N_x M, but I can't realize how to use this information!

 

[Goklayeh]

Maybe, I've solved: let y = x + v, for some (x, v) \in V, and consider any other point p \in M, together with a curve \gamma: (-\epsilon, \epsilon) \to M such that \gamma(0) = x, \dot{\gamma}(0) = v joining x, p. Without loss of generality, y = 0. If f(t):=\frac{1}{2}\left|\gamma(t)\right|^2, then
\dot{f}(0) = \left<\gamma(0), \dot{\gamma}(0)\right> = \left<x, v\right> = 0
since y - r(y) = x + v - x = v \in N_x M. Since f is convex, 0 is a minimum, hence the thesis for the arbitrariness of the curve \gamma. Am I wrong?