Given the set of doubly stochastic matrices of dimension $n$, $D$, is it possible to find a continuous bijective mapping $f: \mathbb{R}^i \to D$ for $i \leq n^2$.
The motivation is to be able to perform gradient descent on the manifold of doubly stochastic matrices.
$D$ is a compact convex set in $\mathbb R^{n \times n}$, and thus a manifold with boundary. Let's say its dimension is $d$. Suppose $f$ existed. The restriction of $f$ to any closed ball is a homeomorphism onto its image. Now using the Baire category theorem, this image must have nonempty interior for some ball, and therefore we must have $i = d$. Now use invariance of domain.