Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function $$ F(v) := \langle \mu \, | \, \xi \mapsto |\xi \cdot v|\rangle $$ defines a norm in $\mathbb{R}^{n+1}$?
Remarks.
1. Note that $\xi \mapsto |\xi \cdot v|$ is not smooth and so a preliminary question asks to determine for what distributions on the $n$-sphere can the function $F$ be defined.
2. Note that if $\mu$ is a signed measure the function $F$ is just the cosine transform of $\mu$
3. For $n = 1$ the solution is that the even part of $\mu$---defined as the average of $\mu$ and its push-forward by the antipodal map---must be a Borel measure whose support contains at least two linearly independent vectors.
I'm actually a bit more interested in the following asymmetric version:
Question 2. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function $$ F(v) := \langle \mu \, | \, \xi \mapsto \max\{\xi \cdot v, 0\}\rangle $$ defines an asymmetric norm in $\mathbb{R}^{n+1}$? Can all asymmetric norms be obtained in this way?
Added. In fact Question 2 is not interesting. This is because the function $\xi \mapsto \max\{\xi \cdot v, 0\}$ equals the average of the functions $\xi \mapsto \xi \cdot v$ and $\xi \mapsto |\xi \cdot v|$. It follows that the only norms that can be obtained from the formula $$ F(v) := \langle \mu \, | \, \xi \mapsto \max\{\xi \cdot v, 0\}\rangle $$ are symmetric norms plus linear forms.
