Second derivative test for intersection bodies
Let \(L\) be an origin-symmetric star body in \(\mathbb{R}^n\). A body \(K\) in \(\mathbb{R}^n\) is called the intersection body of the star body \(L\) if the radial function of \(K\) at every point \(u\) from the unit sphere \(\Omega\) in \(\mathbb{R}^n\) is equal to the \((n-1)\)-dimensional volume of the section of \(L\) by the hyperplane \[ u^\perp= \{x\in\mathbb{R}^n: (x,u)=0\}. \] A body \(K\) in \(\mathbb{R}^n\) is called an intersection body if there exists a finite Borel measure \(\mu\) on \(\Omega\) such that \(\| x\|_K^{-1}=R\mu\), where \(R\mu\) is the finite Borel measure on \(\Omega\) defined by \[ \langle R\mu,f\rangle-\langle\mu, Rf\rangle =\int_\Omega Rf(\theta) d\mu (\theta) \] for every \(f\in C(\Omega)\). \((\| x\|_L\) is defined as \(\| x\|_L= \min \{a\geq 0:x\in aL\}\). The author introduces a test for intersection bodies, in terms of the second derivative of the norm and, by means of this test shows that a large class of bodies in \(\mathbb{R}^n\), where \(n\geq 5\), can be classified as non-intersecting bodies. Several examples are considered: for instance the procedure introduced in this paper enables the author to show that for \(q > 2\) and \(X\) and \(Y\) finite dimensional normed spaces with \(\dim X\geq 1\), \(\dim Y\geq 4\), the unit balls of the \(q\)-sum of \(X\) and \(Y\) is not an intersection body. Similarly, the unit balls of the Orlicz spaces \(\ell^n_M\) where \(n\geq 5\) and the Orlicz function is such that \(M\in C^2([0,\infty]\); \(M'(0)=M''(0)=0\) are not intersection bodies.
scientific article; zbMATH DE number 1215521