simplex – Mathtuition88

Geometric n-simplex is convex

Given the definition of a geometric n-simplex:

$\displaystyle\sigma^n=\{x=\sum_{i=0}^{n}t_i a^i \mid t_i\geq 0\ \text{and }\sum_{i=0}^{n}=1\}\subseteq\mathbb{R}^n$

where $\{a^0,\dots, a^n\}$ are geometrically independent, we can show that the n-simplex is convex (i.e. given any two points, the line connecting them lies in the simplex).

Write $x=\sum_{i=0}^n t_i a^i$ , $y=\sum_{i=0}^n s_i a^i$ .

Consider the line from x to y: $\{ty+(1-t)x\mid 0\leq t\leq 1\}$ .

$\begin{aligned} ty+(1-t)x&=t\sum_{i=0}^n s_i a^i+(1-t)\sum_{i=0}^n t_i a^i\\ &=\sum_{i=0}^n (s_i t+t_i-tt_i)a_i\\ s_it+t_i-tt_i&=s_i t+t_i (1-t)\\ &\geq 0(0)+(0)(1-1)\\ &=0\\ \sum_{i=0}^n s_i t+t_i-tt_i &=t\sum_{i=0}^n s_i+\sum_{i=0}^n t_i -t\sum_{i=0}^n t_i\\ &=t(1)+(1)-t(1)\\ &=1 \end{aligned}$

Thus the line lies inside the simplex, and thus the simplex is convex.

Recommended Books for Math Majors

Video on Simplices and Simplicial Complexes

Professor Wildberger is extremely kind to upload his videos which would be very useful to any Math student studying Topology. Simplices / Simplicial Complexes are usually the first chapter in a Algebraic Topology book.

Check out also Professor Wildberger’s book on Rational Trigonometry, something that is quite novel and a new approach to the subject of Trigonometry. For instance, it can be used for rational parametrisation of a circle.