I am trying to solve this question but am not able to understand how to approach it: What is the polar of an ellipsoid described by the equation:
{$(z_1, . . . , z_d) ∈ R^d: a_1z_1^2 + · · · + a_dz_d^{2} ≤ 1$}, where $a_1, . . . , a_d$ are positive.
I know what generally polar of a set means. Moreover, I am following Algebraic And geometric ideas in the theory of discrete optimization to understand more about the polar of a subset. This link gives a beautiful explanation of how to find polar of various set What are good examples of polar sets in $\mathbb R^2$? I'm trying to approach it the way you solve for a disc but I am still unable to do it. Any help is appreciated
Thank you.
The set $C$ you are considering is
$$ C = \{ (x_1,x_2,\ldots,x_d)\in\mathbb{R}^d \,|\,a_1x_1^2+\cdots + a_dx_d^2 \leq 1\}$$
Then its polar is the set
$$ C^\odot = \{ (x_1,x_2,\ldots,x_d)\in\mathbb{R}^d \,|\,\tfrac{1}{a_1}x_1^2+\cdots + \tfrac{1}{a_d}x_d^2 \leq 1\},$$ which is another ellipsoid. (Note that if $a_1=\cdots=a_d=1$, then you see that $C=C^\odot$ is the unit ball - how beautiful!)
More generally, $$C = \{ x\in\mathbb{R}^d \,|\, \langle x,Qx\rangle \leq 1\} \quad \Rightarrow \quad C^\odot = \{ x\in\mathbb{R}^d \,|\, \langle x,Q^{-1}x\rangle \leq 1\},$$ provided that $Q$ is positive definite.
All this can be found in R.T. Rockafellar's classical Convex Analysis book on page 136.