Positive-spanning poll sets

Let $n\ge 1$ and let $D=(d_1,\dots ,d_m)$ be a nonempty finite ordered list in ${\mathbb{R}}^n$. For coefficients ${\lambda }_1,\dots ,{\lambda }_m$, the finite vector sum ${\sum }_{i=1}^m{\lambda }_i{d}_i$ means the vector whose $k$th coordinate is ${\sum }_{j<m}{\lambda }_{j+1}(d_{j+1}{)}_k$ for each $k<n$; this is the finite real sum of Finite sums and finite products, by recursion. The list is positive spanning if every $w\in {\mathbb{R}}^n$ has such a representation with ${\lambda }_i\ge 0$. Keep the order to make a deterministic poll. Define the directional score ${\psi }_D(v)={\max }_{1\le i\le m}(-⟨v,d_i⟩)$ for unit $v$, and the largest poll norm $D_{\max }={\max }_{1\le i\le m}\mathrm{\parallel }{d}_i{\mathrm{\parallel }}_2$. These maxima exist because their indexing sets are nonempty and finite (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set). A positive uniform lower bound for ${\psi }_D$ is a separate theorem, not part of this definition.