# What is the geometric object $\{x \in \mathbb{R}^n \mid\sum_{i = 1}^n x_i \leq 1, x_i \geq 0\}$ called?

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.

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.
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.