$\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 1}^{n} x_i = 1, x_i\in [0,1]\} = \triangle_{n-1}$, $\overline{v} = (x_1, \cdots, x_n)$ If I will change some parameters, I can have:
$\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 1}^{n} x_i = n, x_i\in [0,n]\}$ -- big $n-1$-simplex, whose vertices are equal to position vectors $n\overline{e_i}$.
$\{\overline{v}\in\mathbb{R}^n \mid x_i\in [0,n]\} = \{0,n\}^n$ -- $n$-hypercube.
$\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 1}^{n} x_i = const, \forall x_i\in \mathbb{R}^1\}$ -- what the "simplex" is this?
$\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 1}^{n} x_i \leq n, x_i\in [-n,n]\}$ something like $n$-dim "rhombus" with the inside whose vertices are equal to position vectors $n\overline{e_i}$ and $-n\overline{e_i}$.
$\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 1}^{n} x_i^2 \leq const^2, \forall x_i\in \mathbb{R}^1\} = D^n_c (0)$ -- $n$-dim disk with center in $0$ and radius $= constant$. But here I am using the euclidean distance and the norm of vectors, right? And if I change metric in space, this condition will not satisfy by properties of disk, yes?
What else can be built with this method of description? Do other sets (excluding item 6) depend explicitly on the space metric?
May be you can give me the name of the textbook, where I can read about this recording form of $\mathbb{R}^n$ subsets?