arXiv subjects
Publications and source records attributed to Damian Pinasco.
We prove that given any set of $n$ unit vectors $\{v_i\}_{i=1}^{n}\subset \mathbb R^n,$ the inequality \[ \sup\limits_{\Vert x \Vert_{\mathbb R^n} =1} \vert \langle x, v_1 \rangle \cdots \langle x, v_n\rangle\vert \ge n^{-n/2} \] holds for $n \le 14.$ Moreover, the equality is attained if and only if $\{v_i\}_{i=1}^{n}$ is an orthonormal system.
We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works when the number of polynomials is large.
For $1< p <2$ we obtain sharp inequalities for the supremum of products of homogeneous polynomials on $L_p(μ)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in the infinite dimensional settings). The results also holds for the Schatten classes $\mathcal S_p$. For $p>2$ we present some estimates on the involved constants.