How do I show that the following sets can be written as the intersection of hyperplanes and half-spaces and hence is a convex set:
$\{p\in\mathbb{R^n}|p_i\in[0,1],\sum_{i=1}^n p_i=1\}$
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$\sum_i p_i = 1$ is a hyperplane. $p_i \ge 0$ is a halfspace, $p_i \le 1$ is also a halfspace. – copper.hat Aug 18 '23 at 16:46
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Your set can be written as:
$$\bigcap_{i=1}^n \left\{p_i \ge 0\right\} \cap \bigcap_{i=1}^n \left\{p_i \le 1\right\} \cap \left\{\sum_{i=1}^n p_i = 1\right\}$$
Do you see what you are looking for?
Kroki
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