Let $E$ be a convex set of $\mathbb{R}^N$, symmetric respect to the origin and bounded. Is it true that there exists an $L^1$ function $l(\theta)$ defined on $S^{N-1}$ with values in $\mathbb{R}^{+}$ such that $$ E=\{ t \cdot \theta : \, 0 \leq t \leq l(\theta) ,\, \theta \in S^{N-1} \}.$$ And what if I consider unbounded sets instead? This fact was crucial for me to prove the following lemma.
Prove that an isoperimetric set $E$ of $\mathbb{R}^n$ which is symmetrical with respect to origin and convex is necessarily a ball. The whole proof stands on proving that the function $l(\theta)$ that I introduced is constant. But I need to prove that $l(\theta)$ admits Lebesgue points first, so I need $l(\theta)$ to be in $L^1(S^{N-1})$ for the Lebesgue differentiation theorem to work, right?