Weak Hadamard Star Configurations and Apolarity

In [7] the authors have introduced a new construction using the Hadamard product to present star configurations of codimension $c$ of ${\mathbb{P}}^n$ and which they called Hadamard star configurations. In this paper, we introduce a more general type of Hadamard star configuration. Any star configuration constructed by our approach is called a weak Hadamard star configuration. We classify weak Hadamard star configurations, and in the case $c=n$, we investigate the existence of a (weak) Hadamard star configuration apolar to the generic homogeneous polynomials of degree $d$.