Operations between sets in geometry

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space $\R^n$. For example, it is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and associative if and only if it is $L_p$ addition for some $1\le p\le\infty$. It is also demonstrated that if $n\ge 2$, an operation * between compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and has the identity property (i.e., $K*\{o\}=K=\{o\}*K$ for all compact convex sets $K$, where $o$ denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. An operation called $M$-addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and GL(n)-covariant operations between compact convex sets in terms of $M$-addition are established. The term "polynomial volume" is introduced for the property of operations * between compact convex or star sets that the volume of $rK*sL$, $r,s\ge 0$, is a polynomial in the variables $r$ and $s$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition

On the large-Q^2 behavior of the pion transition form factor

We study the transition of non-perturbative to perturbative QCD in situations with possible violations of scaling limits. To this end we consider the singly- and doubly-virtual pion transition form factor $\pi^0\to\gamma\gamma$ at all momentum scales of symmetric and asymmetric photon momenta within the Dyson-Schwinger/Bethe-Salpeter approach. For the doubly virtual form factor we find good agreement with perturbative asymptotic scaling laws. For the singly-virtual form factor our results agree with the Belle data. At very large off-shell photon momenta we identify a mechanism that introduces quantitative modifications to Efremov-Radyushkin-Brodsky-Lepage scaling.Comment: 5 pages, 7 figures, v3:contents revised, version published in PL