Volume of a shape in N dimensions defined by an inequality.

Question

What does the inequality $x_1+x_2+x_3+...+x_N < E$ represent in an $N$-dimensional space, and what is its volume?

I tried to extrapolate from 3-dimensional space, in which we have, let's say, $x_1+x_2+x_3 < 10$. Then I decomposed this inequality into 3 inequalities as $x_1+x_2<\dfrac{20}{3}$, $x_1+x_3<\dfrac{20}{3}$, and $x_2+x_3<\dfrac{20}{3}$. I know this should correspond to a tetrahedron with 3 rectangles, and the volume of it should be $\dfrac 1 2 \left(\dfrac{20}{3}\right)^3$

But what is the case of $N$-dimensional space?

Answer 1

The inequality defines a so-called simplex. Two weeks ago, I answered a similar question; take a look at my answer.