We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$, we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$.

Compared to studies of the effects of formal training in translation, little is known about the psycholinguistic impact of the experience of informal translation, or language brokering. The present study examined this issue in the context of idiom comprehension. Bilingual adults differing in prior brokering experience read English idioms and judged whether target words presented in English or Spanish were related to the idiom's meaning. For brokers, relatedness judgments were not affected by whether the targets were in the same or different language as the idiom; however, non-brokers were faster for same-language than different-language idiom-target pairings. The findings suggest that language brokering experience facilitates idiom meaning comprehension even across language boundaries, with further differences related to idiom decomposability. More generally, the findings underscore the importance of considering systematic sources of variability in language practice among bilinguals, aside from differences related to proficiency, in theorizing effects associated with bilingualism.

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.