According to references (e.g. Wikipedia and elsewhere), the Dirichlet distribution, parametrized by $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_K)$, is $$ D(x_1, \ldots, x_K) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1} $$ where $$ \mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^K \alpha_i\right)}. $$ So, if $K = 2$ and $\alpha_1 = \alpha_2 = 1$ then this gives $ D(x_1, x_2) = 1/\mathrm{B(\boldsymbol\alpha)} $ where $$ \mathrm{B}(\boldsymbol\alpha) = \Gamma(1)^2 / \Gamma(2) = 1 $$ so, $D(x_1, x_2) = 1$ for all $x_1, x_2$. However, $D(x_1, x_2)$ is defined on the standard $1$-simplex defined in $R^2$ by $x_i \ge 0$ and $x_1 + x_2 = 1$. This is the span (or affine hull) of the two points $(0, 1)$ and $(1, 0)$. Since this is a line segment of length $\sqrt{2}$, the integral of the Dirichlet distribution over this simplex is $\sqrt{2}$, not $1$ as expected. What am I missing here?
The same problem comes in higher dimensions. For instance, for $K=3$, the simplex is a triangle with side $\sqrt{2}$, but the normalization constant becomes $B(\boldsymbol\alpha) = 1/\Gamma(3) = 1/2$, which is not the area of this triangle.
What is wrong here?