What the set :
$$S = \{ (\lambda_1, ..., \lambda_n) \in \mathbb{R}_+^n \mid \sum \lambda_i = 1 \}$$
represent geometrically ? I tried in dimension $2$ and it seems that I get a triangle. But I can't see what it represents in higher dimension. Moreover we have $S \subset \mathbb{S}^{n-1}$.
Moreover I am wondering how the shape changes if we only allowed the $\lambda_i > 0$ (and not $\lambda_i \geq 0$ ). That's to say how does this set :
$$\{ (\lambda_1, ..., \lambda_n) \in \mathbb{R}_+^{*n} \mid \sum \lambda_i = 1 \}$$
differs from the set $S$ geometrically ?
Thank you !
If you have the geometry of the unit sphere of $\mathbb{R}^n$ for the $1$-norm in mind :
$$\|x\|_1 = \sum |x_i|$$
Then $S$ is just the intersection of that sphere and $\mathbb{R}^n_+$.
Hence for $n=2$, $S$ is the line joining the points $(1,0)$ and $(0,1)$. For $n=3$ it is the (full) triangle with vertices $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$.
In that case, the difference between $S$ and the second set is that you just have to remove the points $(0,1)$ and $(1,0)$ of the line joining them. For $n=3$, you remove the edges of the (full) triangle, and so on.
From the point of view of convex sets, $S$ is the convex hull of the points of the form $(0,\dots,0,1,0,\dots,0) \in \mathbb{R}^n$ (only one component is $1$ and the others are $0$). Note that they are all contained in a hyperplane.