An application of John ellipsoids to the Szego kernel on unbounded convex domains

We use convex geometry tools, in particular John ellipsoids, to obtain a size estimate for the Szeg\H{o} kernel on the boundary of a class of unbounded convex domains in ${\mathbb{C}}^n\mathrm{.}$ Given a polynomial $b:$\mathbb{R}$^n \rightarrow $\mathbb{R}$$ satisfying a certain growth condition, we consider domains of the type $\Omega_b = { z\in$\mathbb{C}$^{n+1},:, {\rm Im}[z_{n+1}]

b({\rm Re}[z_1],\ldots,{\rm Re}[z_n]) }.$