The polar dual of the $\ell^p,p\in\Bbb N$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ is

$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$

with $q=p/(p-1)$. Using Tarski-Seidenberg we see that the polar dual of a semi-algebraic set is semi-algebraic (see e.g. the discussion here). But if $p\ge 3$, then $q\not\in\Bbb N$, hence $(*)$ is not a semi-algebraic description.

Question: Is there a nice semi-algebraic description of $(*)$?

I am mostly interested in $n=2$.

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is

$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$

with $q=p/(p-1)$. Using Tarski-Seidenberg we see that the polar dual of a semi-algebraic set is semi-algebraic (see e.g. the discussion here). But if $p\ge 3$ then $q\not\in\Bbb N$, hence $(*)$ is not a semi-algebraic description.

Question: Is there a nice semi-algebraic description of $(*)$?

I am mostly interested in $n=2$.

The polar dual of the $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in \Bbb N$ is

$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$

with $q=p/(p-1)$. Using Tarski-Seidenberg we see that the polar dual of a semi-algebraic set is semi-algebraic (see e.g. the discussion here). But if $p\ge 3$, then $q\not\in\Bbb N$, hence $(*)$ is not a semi-algebraic description.

Question: Is there a nice semi-algebraic description of $(*)$?

I am mostly interested in $n=2$.

The polar dual of the $\ell^p,p\in\Bbb N$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ is

$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$

with $q=p/(p-1)$. Using Tarski-Seidenberg we see that the polar dual of a semi-algebraic set is semi-algebraic (see e.g. the discussion here). But if $p\ge 3$, then $q\not\in\Bbb N$, hence $(*)$ is not a semi-algebraic description.

Question: Is there a nice semi-algebraic description of $(*)$?

I am mostly interested in $n=2$.