Let $d \in \mathbb{N}, 1\leq p \leq \infty$, $f \in C^{\infty}_0 (\mathbb{R}^d)$.
It is known that
there is a function $g$ such that
$g(x)=g_0(|x|)$ for some non-increase function $g_0:[0, \infty) \to \mathbb{R}$,
$||f||_{L^{p} (\mathbb{R}^d)}=||g||_{L^{p} (\mathbb{R}^d)}$, and
$||\nabla f||_{L^{p} (\mathbb{R}^d)} \geq|| \nabla g||_{L^{p} (\mathbb{R}^d)}$.
Foe example, the symmetric decreasing rearrangement $g=f^*$ satisfies this, and
$||\nabla f||_{L^{p} (\mathbb{R}^d)} \geq|| \nabla f^*||_{L^{p} (\mathbb{R}^d)}$ is called the Pólya–Szegő inequality.
I'm interested in generalization of this for $p=1$:
Let $K \subset \mathbb{R}^d$ be a nonempty compact set such that
the Lebesgue measure $|K|=0$.
Does there exist a function $h$ such that
$h(x)=h_0(d(x,K))$ for some non-increase function $h_0:[0, \infty) \to \mathbb{R}$ ($\displaystyle d(x,K)= \inf_{y \in K} |x-y|$),
$||f||_{L^{1} (\mathbb{R}^d)}=||h||_{L^{1} (\mathbb{R}^d)}$, and
$||\nabla f||_{L^{1} (\mathbb{R}^d)} \geq|| \nabla h||_{L^{1} (\mathbb{R}^d)}$?
Any advise would be appreciated.