Hölder Continuity for the Inverse of Mañé′s Projection

In a celebrated work, Mañé showed that given a compact set X with fractal dimension d in a Banach space B, there exists a projection P of rank ≤2d + 1, such that P restricted to X is injective. Here, we prove a stronger result when B is finite dimensional, namely, that there exists an orthogonal projection P0 such that not only P0 is injective on X but also its inverse is Hölder continuous when restricted to P0X.