It is my first post, so please correct me if I am not following the rules/etiquette.
Assume that we are given a space $\mathcal{S}$ composed by vectors $x\in\mathbb{R}^L$, constrained by $$\sum x_i=1,\\ 0\le x_i \le 1,$$ where $x_i$ is the $i$-th element of the vector $x$. We also assume to be able to measure the distance between the elements, $$d(x,y)=\sqrt{x^TAy},$$ where $A$ is a given positive semi-definite matrix.
If I stand correct, we cannot define a vector space because of the constraints on the vectors, i.e. the sum of two element in $\mathcal{S}$ does not belong to $\mathcal{S}$.
I would like to compute quantities related to a set of elements belonging to $\mathcal{S}$, for example the mean and the variance. We define such set as $\{y^n\}_{n=1}^N\text{ where } y^n\in\mathcal{S}$.
I thought of considering the Frechet mean/variance and solve the following optimization problem:
$$ \min_{x \in \mathcal{S}} \sum_{n=1}^N d^2(x,y^n) = \min_{x\in\mathcal{S}} \sum_{n=1}^N x^TAy^n=\min_{x\in\mathcal{S}}x^TA\bar{y},$$ where $\bar{y} = \sum_{n=1}^N y^n$.
This looks like an LP program that could be easily solved. The minimizer would be the Frechet mean and the argument would be the Frechet variance.
It looks ok to me, but I am afraid that I have missed some details around that could not allow the solution of the optimization problem.
1) Is this the approach correct? 2) I could not find much information by searching around and I was wondering if there were some pointers around for similar works.