Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains

Let $Ω\subset\mathbb{R}^N$ be a bounded convex domain and let $A=-Δ_D$ be the positive Dirichlet Laplacian. For every real $s>0$ with $N>2s$, we prove that the function \[ (x,y)\longmapsto K_{s,Ω}(x,y)^{-1/(N-2s)}, \] where $K_{s,Ω}$ is the Green kernel of $A^{-s}$, extends continuously by zero to