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Is there a name for the geometric object

$$C = \left\{x \in \mathbb{R}^n \,\,\Big\lvert\, \sum\limits_{i = 1}^n x_i \leq 1, x_i \geq 0\right\}\quad?$$

It looks like a filled triangle in 2D.

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As noted by cmk, what you gave is a nonstandard representation of a simplex, in particular an $n$-simplex. The $n$-simplex is typically represented in $\mathbb{R}^{n+1}$ by $$ \left\{ x \in \mathbb{R}^{n+1} : \sum_{i=1}^{n+1} x_{i} = 1 \ \mathrm{and} \ x_{i} \geq 0 \ \mathrm{for} \ \mathrm{all} \ i \right\} $$ but as noted in the wikipedia article under the "Corner of cube" section it is possible to represent the $n$-simplex in $\mathbb{R}^{n}$ as the corner of the $n$-cube using exactly the definition you gave. It's a neat subtlety that both constructions give the same polytope. The reason it works is that from any $n$ coordinates $x_{i}$ summing to less than one, we can always define an additional $$x_{n+1} = 1 - \sum_{i=1}^{n} x_{i}$$ satisfying $x_{n+1} \geq 0$ and by definition we then have $$ \sum_{i=1}^{n+1} x_{i} = 1.$$ Conversely, if we $n+1$ coordinates, we can just forget about that last one. This gives a 1-1 correspondence between the two representations.

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That is a simplex. Usually the dimension is included in the name and that object is called a "$k$-simplex". (link: Simplex on Wikipedia)

A $k$-Simplex lives in dimension $k$, i.e. the 2-simplex is a triangle, as you observed.

The answer from cmk is nearly correct, but the dimension is one off.

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