Plank type problems for polynomials on Banach spaces
Let \(E\) be a Banach space over \(\mathbb K\) (\({\mathbb K}={\mathbb R}\) or \(\mathbb C\)) and \(n\) be a positive integer. A (continuous) mapping \(P: E\to {\mathbb K}\) is said to be an \(n\)-homogeneous polynomial if there is a (continuous) symmetric \(n\)-linear mapping \({\buildrel\vee\over P}: E\times E\times\dots\times E \to {\mathbb K}\) such that \(P(x)={\buildrel\vee\over P}(x,\dots,x)\) for all \(x\) in \(E\). The norm of \(P\) is given by \(\|P\|=\sup_{\|x\|\leq 1}|P(x)|\). The space of all \(n\)-homogeneous polynomials on \(E\) is denoted by \({\mathcal P}(^nE)\). Given a Banach space \(E\) and natural numbers \(n\) and \(k\), the authors introduce the plank-type constant for homogeneous polynomials \(c(n,k,E)\), as \(c(n,k,E)=\sup\{c : \text{for all norm one } P_1, P_2, \dots , P_n \text{ in } {\mathcal P}(^kE) \text{ there is } x\in S_E \text{ such that } |P_j(x)|\geq c, ~j=1,\dots , n\}\). They show that \(\displaystyle{\left({1\over n}\right)^{nk}\leq c(n,k,E)}\) for all \(n\), \(k\) and \(E\), that \(c(n,k,E)\) is a decreasing function of both \(n\) and \(k\) and that \(c(n,k,E)<1\) when \(\dim(E), n\geq 2\), \(k\geq 1\), and \(E'\) is strictly convex. When \(H\) is a real Hilbert space it is shown that \(\displaystyle{ c(2,2,H)={1\over 3}}\). This result enables the authors to give lower bounds for \(c(2,2,L^p(\mu))\), \(p\geq 1\), and \(c(2,2,E)\) for any real Banach space \(E\).
scientific article; zbMATH DE number 6096048
scientific article