I am curious about studying the following subset $X$ of $\mathbb{R}^n$: $$X:=\{(x_1,\cdots,x_n)\in\mathbb{R}^n|x_i\ge 0\ \mathrm{for}\ \forall i,\sum_{i=1}^nx_i=1\}.$$ I can see that $X$ is significantly harder to handle mathematically than ordinary vector subspaces of $\mathbb{R}^n$, because $X$ is not closed under addition or scalar multiplication commonly defined in $\mathbb{R}^n$ ($X$ does not even contain an additive identity in $\mathbb{R}^n$, which is $0$). Because of the restriction $x_i\ge 0$, we cannot directly study $X$ in terms of elementary linear algebra, although one could still formally apply linear transformation defined in $\mathbb{R}^n$ with some restrictions (if this is not the case, please let me know).
Nevertheless, it seems to me that $X$ still has some nice properties, be it geometrically or algebraically (e.g. flatness, symmetry with respect to permutation of indices, ...), which makes me suspect that some theory can be built upon sets like $X$.
Since $X$ or automorphisms on $X$ (possibly with some suitable topology introduced), or more generally algebra on "flat (maybe closed) halfspace" can be of interest in several application contexts, I expect that there has already been considerable amount of literature dealing with this kind of situations, but I still haven't been aware of any textbooks or articles, or I have no idea what is the suitable keywords for Google. So I would be grateful if you kindly suggested me what to read first.