Generalized Levy representation of norms and isometric embeddings into \(L_ p\)-spaces (Q1201363)

The author proves that if \(p>0\), \(p\neq 2,4,6,\dots\) then the norm of every \(n\)-dimensional Banach space admits a representation \[ \| x\|^ p=\int_{\mathbb{R}^ N}|\langle x,s\rangle|^ pd\gamma(s), \] where \(\gamma\) is a distribution. This representation is used in order to obtain criteria for isometric embeddability of Banach spaces into \(L_ p\) spaces.