# Prove K is a convex body and find the polar of K (K°)

Say $$K$$ = { x$$R^3$$ : $$\frac{x_1^2}{a_1}$$ + $$\frac{x_2^2}{a_2}$$ + $$\frac{x_3^2}{a_3}$$ $$\leq$$ 1 }. Show that K is a convex body. I've managed to show that K is centrally symmetric and closed. How can I prove that it's convex and bounded above and below, exam style?

Also, how do I go about finding the polar of $$K$$ for a set like this. I've started by finding the dot product of $$x$$ and $$y$$ in $$R^3$$.

So xy = $$x_1y_1$$ + $$x_2y_2$$ + $$x_3y_3$$ and I know I should be using Cauchy Schwartz at some point but where do I go from here?