Here is the result on linear forms :
Let consider $\psi,\psi_1,...,\psi_p$, $\ p+1$ linear forms on $E$ a $\mathbb{K}$-vector space of finite or infinite dimension. Then the following conditions are equivalent :
$\bullet $$\bigcap\limits_{i=1}^p \ker(\psi_i)\subset \ker(\psi)$.
$\bullet$ $\psi \in \text{span}\{\psi_1,...,\psi_p\}$.
Now here is the statement :
Let consider the Euclidean space $\mathbb{R}^n$. Let $x,x_1,...x_p$ be vectors of $\mathbb{R}^n$. Then the following conditions are equivalent :
$\bullet$ $x\in \text{span}\{x_1,...,x_p\}$ and the coefficients of the linear combination are nonnegative.
$\bullet$ for all $h\in \mathbb{R}^n$ such that $\forall i \in \{1,...,p\}, \ \langle h,x_i\rangle \ge 0$ we also have $\langle h,x\rangle\ge 0$.
Moreover if someone has references about it and these kind of results, it would be great !
Thanks in advance !