how to find the polar of a set:$A=\{x\in R_n:\sum (λ_ix_i)^2\leq 1\}$, where $ λ_i $is any constant Can you give some details about how to get the polar of $A=\{x\in R_n:\sum (λ_ix_i)^2\leq 1\}$, where $λ_i$ is any constant; I barely have any clue about this. Thanks!

The key is the Cauchy-Schwarz inequality:

For all $a,b \in \mathbb{R}^n$, you have $\sum_{i=1}^n a_i \, b_i \le \|a\|_2 \, \|b\|_2$. Equality holds iff $\alpha \, a = \beta \, b$ for some $\alpha, \beta \ge 0$.

Try to apply this to the vectors with components $a_i = \lambda_i \, x_i$ and $b_i = \lambda_i^{-1} \, y_i$. 
