An ellipse $E$ can be defined by two foci, $p,q\in\mathbb{R}^2$, and a length parameter $\ell$ as follows: $$ E = \{x\in\mathbb{R}^2 : ||p-x||+||q-x||\le\ell \}.$$ The area of $E$ is uniquely determined (and easily calculated) as a function of $||p-q||$ and $\ell$.
Let us now define a $2$-hop ellipse $E_2$ as follows: $$ E_2 = \{(x_1,x_2)\in(\mathbb{R}^2)^2 : ||p-x_1||+||x_1-x_2||+||q-x_2||\le\ell \}.$$
Is anything known about the 4-dimensional Lebesgue measure of $E_2$ --- which is uniquely determined by $||p-q||$ and $\ell$? What about the natural generalizations to $k$-hop ellipses?
