How to show that $X \ast Y$ is convex?

Question

Let $X, Y \subset \mathbb{R}^{n}$ and $X \ast Y$ be the union of all straight lines $[x, y]$ where $x \in X$ and $y \in Y$. Suppose that $X, Y$ are convex sets, show that $X \ast Y$ is a convex set.

I don't know how to work with this set to show that it is convex. A straight line in $X$ can be written as $$[x,y] = \lbrace (1-t)x + ty\;|\;0 \leq t \leq 1 \rbrace.$$ How do I do this in $X \ast Y$? $$\left[[x,y],[x',y']\right] = \lbrace (1-t)[x,y] + t[x',y']\;|\; 0 \leq t \leq 1\rbrace ?$$ Thanks for any hints!

Answer 1

Defining

$s(x_1,y_1,\lambda_1) = \lambda_1 x_1 + (1 - \lambda_1) y_1$

$s(x_2,y_2,\lambda_2) = \lambda_2 x_2 + (1 - \lambda_2) y_2$

with $\lambda_1 \in [0,1]$ and $\lambda_2 \in [0,1]$

then $s(x_1,y_1,\lambda_1), s(x_2,y_2,\lambda_2)$ both are elements of $X * Y$

To demonstrate convexity we shall prove that $\mu s_1 + (1-\mu)s_2 \in X * Y$ with $\mu \in [0,1]$

so

$s(\mu x_1+(1-\mu)x_2,\mu y_1+(1-\mu)y_2,\lambda) = \mu s(x_1,y_1,\lambda)+(1-\mu)s(x_2,y_2,\lambda) \in X * Y$

Answer 2

Hint: Let $z_1,z_2\in X\ast Y$, i.e. $z_i\in [x_i, y_i]$ for some $x_1,x_2\in X, \ y_1,y_2\in Y$.
Then express any $z=tz_1+(1-t) z_2$ using $x:=tx_1+(1-t)x_2\, \in X$ and $y:=ty_1+(1-t)y_2\, \in Y$.