The isoperimetric quotient and some classical Banach spaces (Q1812764)

Let \(E\) be a normed space, naturally identified with \(\mathbb{R}^ n\) and the unit ball \(B_ E\). Write \(P_ \xi(B_ E)\) for the orthogonal projection of \(B_ E\) onto the \((n-1)\)-dimensional subspace with normal \(\xi\). The first of the two principal results is an upper bound for \(\rho=\max_{\xi,\eta}(v_{n-1}(P_ \xi(B_ E))/v_{n-1}(P_ n(B_ E))\) in terms of the isoperimetric quotient \(\kappa=\nu_{n-1}(\partial B_ E)/(v_ n(B_ E))^{(n-1)/n}\), where \(v_ n\), \(v_{n-1}\) signifies the volume of the indicated dimension, and certain restrictive assumptions are made on a basis for \(E\). In particular, if \(E\) has a basis \(\{e_ i\}\), \(i=1,\dots,n\), such that \(\|\sum_{i=1}^ n a_ i e_ i\|=\|\sum_{i=1}^ n a_{\pi(i)}\varepsilon_ i e_ i\|\) for all permutations \(\pi\) of \((1,\dots,n)\) and for all choices of \(\varepsilon_ i=\pm1\), then there is an absolute constant \(C\) such that \(C^{-1}\kappa/\sqrt{n}\leq\rho\leq C\kappa/\sqrt{n}\). His second principal result bounds quotients of the type \(\sigma(L)=v_{n-1}(P_ \xi(B_ L))/v_{n-1}(\partial(B_ L))\), where \(L\) is a Lorentz space; for example, the author shows that if \(L=\ell_{p,q}^ n\) with norm \(\| x\|_{p,q}=(\sum_{i=1}^ n i^{q/(p-1)}| x_ i^*|^ q)^{1/q}\), where \(| x_ i^*|\) is the decreasing rearrangement \(q| x_ i|\) and if \(\max(1,p/2)<q\leq p\leq\infty\) then there is a constant \(c_{p,q}\) such that \(c_{p,q}^{-1}/\sqrt{n}\leq\sigma(\ell_{p,q}^ n)\leq c_{p,q}/\sqrt{n}\).