Let $\kappa:\mathbb{R}^d\times \mathbb{R}^d \to\mathbb{R}$ be a positive semidefinite kernel. If $A,B \subseteq \mathbb{R}^d$ are disjoint sets of equal, finite measure, then applying the definition of positive semidefiniteness to estimate the inner product of $\mathbf{1}_A-\mathbf{1}_B$ with its convolution with $\kappa$ shows that $$ \frac{1}{|A\cup B|^2}\iint_{A\cup B} \kappa(x,y)dxdy \leq \frac{1}{2}\left[\frac{1}{|A|^2}\iint_{A} \kappa(x,y)dxdy + \frac{1}{|B|^2}\iint_{B} \kappa(x,y)dxdy\right], $$ where $|A|$ denotes the volume of $A$. This can be thought of as a kind of monotonicity for the average of $\kappa$ over a set. Applying this estimate inductively yields that, if $\kappa(x,y)=\kappa(0,y-x)$ is translation-invariant, then the average of $\kappa$ over a box $[0,kr]^d$ is at most the average over $[0,r]^d$ for every $r>0$ and every natural number $k\geq 1$. (To see this, use the above fact to bound the average over $[0,kr]^{\ell}\times[0,r]^{d-\ell}$ inductively for $\ell=0,1,\ldots,d$.)
I am curious about the following questions:
- Is there a standard name or reference for this fact about averages of positive-definite functions over boxes decreasing in the side-length?
- Is there an analogous statement for balls, rather than boxes? What about other convex sets? (Of course we can get coarse versions of the inequality, involving a set-dependent constant, by taking boxes that inscribe and circumscribe the given convex set.)
- To what extent can the assumption that the scaling factor is an integer be relaxed?
