What is a valid mapping from the simplex to positive orthant of sphere?

Question

Where the simplex is all points such that the sum of entries is 1, and the sphere is all points such that the squared sum of entries is 1.

Where the norm is given by Fisher metric:

$$||\dot X_t||_{X_t}^2 = \sum_{i=1}^n (\dot X_t)_i^2 / (X_t)_i$$

To find mapping from: $$\Delta = \left\{x \Bigg| x_i \ge 0, \sum_{i=1}^n x_i = 1\right\}$$

To the positive orthant of the sphere: $$S = \left\{y \Bigg| y_i \ge 0, \sum_{i=1}^n y_i^2 = 1\right\}$$

Answer 1

Hint: divide by the square root of the sum of squares. Explicitly, define $f:\Delta \to S$ by $$f(x_1,\dots,x_n)=\frac{(x_1,\dots,x_n)}{\sqrt{\sum_{i=1}^n x_i^2}}.$$ Now show that $f(x_1,\dots,x_n) \in S$.

Answer 2

Let me give the correct answer: scale by the euclidean norm.

The unit sphere is just the set of vectors of unit length. So enough to scales vectors by their lengths

$$f(x) = x / \|x\|_2,$$

recall that $\|x\|_2 = \sqrt{\sum_{i=1}^{n} x_i^2}$.