Is the Sum of Two Elements in a Convex Compact Subset of R^2 Also in the Subset?

[almostsure]

Let $X$ be a compact and convex subset of $\mathbb{R^2}$

Let $a^1, a^2 \in X$ such that $a^j = (a^j_1, a^j_2)$, $j=1,2$

Is $c= \sum_{i=1}^2 \mathbb{I}_{ i=j} a^j_i \in X \quad ?$

 

[Stephen Tashi]

The '$' won't work as a tag to encapsulate LaTex on this forum. See https://www.physicsforums.com/showthread.php?t=546968

I don't understand all the symbolism in your question, but using the tags

 [itex]...[/itex]

instead of $, it comes out as:

Let $X$ be a compact and convex subset of $\mathbb{R^2}$

Let $a^1, a^2 \in X$ such that $a^j = (a^j_1, a^j_2)$,$j=1,2$

Is $c= \sum_{i=1}^2 \mathbb{I}_{ i=j} a^j_i \in X \quad ?$