I have a simple question about the definition of a simplex. In this paper by Jonathan Huang he notes that every point on the simplex can be expresed as a linear combination $$x=\sum\limits_{i=0}^n \alpha_i p_i$$ with $\sum\limits_{i=0}^n \alpha_i \leq 1$ and $\alpha_i\geq 0$. However, on Hatcher's book (page 7) or on Wikipedia, for example, it is defined with $\sum\limits_{i=0}^n \alpha_i =1$.
What's the most accepted definition?