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_1$$y_1$ + $x_2$$y_2$ + $x_3$$y_3$ and I know I should be using Cauchy Schwartz at some point but where do I go from here?


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