Let $v=(v_1,\cdots,v_n)\in\mathbb R^n$ and $\Lambda\subset \mathbb R^n$ ($\Lambda$ is a discrete set) with $v\in\Lambda$. Consider the subset of $\mathbb R^n$ given by the set of those $x\in \mathbb R^n$ for which $\sum v_ix_i=|v|^2/2$; and for all $a=(a_1,\dots,a_n)\in \Lambda$, we have $\sum a_ix_i\le |a|^2/2$. Can we get a necessary and sufficient condition on $v$ for which this will be an affine subspace of dimension $n-1$?
For reasons outlined in this paper I believe that the condition must be $v^Ta<a^Ta$ for all $a\in\Lambda-\{0,v\}$. But there seems to be some easy reason apparently for this which I am unable to identify.