Are polynomials with positive coefficients convex in $[0,1]^d$?

Question

Are polynomials of a given order with positive coefficients, for example

$$a+bx_1+cx_2+dx_3+ex_1x_2+fx_1^2x_3+gx_2^3$$

convex in $(x_1,x_2,x_3) \in[0,1]^3$? If not, is it convex in $1'x \le 1$ and $x \ge 0$?

Answer 1

Convexity requires the Hessian matrix to be positive semidefinite. The Hessian matrix of the quadratic polynomial $$ Ax_1^2 + 2Bx_1x_2 + Cx_2^2 $$ is precisely $$\begin{pmatrix} A & B \\ B & C\end{pmatrix}$$ at every point $(x_1, x_2)$.

Is every symmetric matrix with positive entries positive-semidefinite? No, a counterexample being $$\begin{pmatrix} 1 & 2 \\ 2 & 1\end{pmatrix}$$ which corresponds to $$ x_1^2 + 4x_1x_2 + x_2^2 = (x_1+x_2)^2 + 2x_1x_2 $$ The lack of convexity can be seen directly by restricting this polynomial to the line $x_2 = \alpha - x_1$.