0

Suppose we have a set $X=\{1,...,n\}$ where $n$ is a natural number.

Let $\Delta(X)$ be the set of all probability distributions over $X$.

Then $\Delta(X)$ is a convex set.

How do we interpret convexity in this case? What would be sufficient to show to prove that $\Delta(X)$ is a convex set?

I know that the elements of $\Delta(X)$ satisfy two properties, namely: non-negativity and the sum of all the elements is equal to $1$. But I am unsure how to start proving convexity.

Michael
  • 23,905
johnny09
  • 1,535
  • 2
    You can define $$\Delta(X) = \left{(p_1, ..., p_n) \in \mathbb{R}^n : \sum_{i=1}^n p_i=1, p_i \geq 0 : \forall i \in {1, ..., n}\right}$$ So $\Delta(X) \subseteq \mathbb{R}^n$. Then use the definition of a convex set. Do you know the definition? – Michael Sep 19 '18 at 04:04
  • @Michael I am not familiar with the definition of a convex set and although I have searched in the internet, I am not sure which one is appropriate. Could you please direct me to a good source? – johnny09 Sep 19 '18 at 04:15
  • 1
    Def: A set $\mathcal{A}$ is a convex set if for all $\vec{x},\vec{y} \in \mathcal{A}$ and all $\theta \in [0,1]$ we have $$\theta \vec{x} + (1-\theta) \vec{y} \in \mathcal{A}$$ – Michael Sep 19 '18 at 04:19

0 Answers0