The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the inequalities from the system. The precise statement is as follows:

Let $L_1,\dotsc,L_m$ and $P$ be linear polynomials in the $n$-dimensional real variable $x=(x_1,\dotsc,x_n)$, and suppose that the set of all those $x$ with $L_1(x)\ge 0,\dotsc,L_m(x)\ge 0$ is non-empty. If $P(x)\ge 0$ for each $x$ from this set, then there exist $c_1\ge 0,\dotsc,c_m\ge 0$ with $P\ge cL_1+\dotsb+cL_m$.

For $P$ quadratic this may fail: consider, for instance, $L_1(x)=x$, $L_2(x)=1-x$, and $P(x)=x(1-x)$. I wonder, however, whether the assertion stays true if we allow summands of the form $L_iL_j$:

Suppose that $L_1,\dotsc,L_m$ are linear, and $P$ a quadratic polynomial in the $n$-dimensional real variable $x=(x_1,\dotsc,x_n)$. Given that $P(x)\ge 0$ whenever $L_1(x)\ge 0,\ldots,L_m(x)\ge 0$ (and the set of all such $x$ is non-empty), must there exist $c_i,c_{ij}\ge 0$ with $P\ge \sum c_iL_i+\sum c_{ij} L_iL_j$?

I was able to settle some particular cases; most notably, that where $n=1$ (one variable), and also that where $m=1$ (one constraint). Perhaps, with some effort I can also resolve the case $m=n=2$ (from which the case of $m=2$ and $n$ arbitrary will follow, if I am not mistaken).

I would expect that this should have been known; can anybody suggest a reference?

The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the inequalities from the system. The precise statement is as follows:

Let $L_1,\dotsc,L_m$ and $P$ be linear polynomials in the $n$-dimensional real variable $x=(x_1,\dotsc,x_n)$, and suppose that the set of all those $x$ with $L_1(x)\ge 0,\dotsc,L_m(x)\ge 0$ is non-empty. If $P(x)\ge 0$ for each $x$ from this set, then there exist $c_1\ge 0,\dotsc,c_m\ge 0$ with $P\ge cL_1+\dotsb+cL_m$.

For $P$ quadratic this may fail: consider, for instance, $L_1(x)=x$, $L_2(x)=1-x$, and $P(x)=x(1-x)$. I wonder, however, whether the assertion stays true if we allow summands of the form $L_iL_j$:

Suppose that $L_1,\dotsc,L_m$ are linear, and $P$ a quadratic polynomial in the $n$-dimensional real variable $x=(x_1,\dotsc,x_n)$. Given that $P(x)\ge 0$ whenever $L_1(x)\ge 0,\ldots,L_m(x)\ge 0$ (and the set of all such $x$ is non-empty), must there exist $c_i,c_{ij}\ge 0$ with $P\ge \sum c_iL_i+\sum c_{ij} L_iL_j$?

I was able to settle some particular cases; most notably, that where $n=1$ (one variable), and also that where $m=1$ (one constraint). Perhaps, with some effort I can also resolve the case $m=n=2$ (from which the case of $m=2$ and $n$ arbitrary will follow, if I am not mistaken).

I would expect that this is either false, or should be known. Can anybody construct a counterexample or suggest a reference?