Let $X,Y\subset \mathbb{R}^n$ are compact and convex. Define $S=\{\langle x,y\rangle\in \mathbb{R}\mid x\in X, y\in Y\} $. Is $S$ convex?
I tried to prove by taking an arbitrary convex combination of two elements in $S$ and tried to show it is also in $S$ but I got nowhere. I don't even know if the statement is true. Is it true? If so, any hint on how to prove it would be greatly appreciated.
It is true, and it suffices that $X,Y\subset \mathbb{R}^n$ are connected. (This is more general since all convex sets in $\Bbb R^n$ are connected.)
Then $X \times Y$ is also connected (see e.g. here) and the image of $X \times Y$ under the continuous function $f(x, y) = \langle x,y\rangle$ is a connected subset of $\Bbb R$ (see e.g. here).
The connected subsets of $\Bbb R$ are exactly the intervals, and intervals are convex sets.
If, in addition, both $X$ and $Y$ are compact then $S$ is a compact interval, i.e. $S = [a, b]$ with real number $a \le b$.