Checking the convexity of a quadratic function with bilinear terms

Question

I have a function of the following form

$$a(b(x^2 +y^2+z^2 +xy + xz + yz)+ x+y+z)$$

where $a, b > 0$. Is this function convex? How can we show convexity? Thank you.

Answer 1

The function $(x,y,z) \mapsto x+y+z$ is linear hence convex.

The Hessian of $(x,y,z) \mapsto x^2+y^2+z^2+xy+yz+xz$ is $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ which is symmetric, real and has eigenvalues $1,1,4$.

If $f$ is convex and $b > 0$ then $b \cdot f$ is convex.

If $g,h$ are convex and $a >0$ then $a(g+h)$ is convex.