Osteoporosis-related fractures have a major impact on health at the individual and societal levels, through associated morbidity and increased mortality. Up to 50% of women and 20% of men at age 50 years may have a fragility fracture in their remaining lifetimes. Nutrition is important throughout the life course. Thus, adequate Ca and vitamin D intake has been shown to reduce risk of fracture in old age. Other factors such as protein and vitamin K may also be important, although the evidence here is less strong. In childhood Ca or vitamin D supplementation trials have demonstrated modest short-term increases in bone mass, but the long-term implications have not been established. Over recent years it has become apparent that maternal nutrition may have critical and far-reaching persistent consequences for offspring health. Thus, reduced maternal fat stores and low levels of circulating 25-hydroxyvitamin D in pregnancy are associated with reduced bone mass in the offspring; placental Ca transport may be key to these relationships. Wider maternal dietary patterns have also been shown to predict offspring bone mass. These data suggest that an interventional approach aimed at specific micronutrients, such as vitamin D, should be complemented by general optimisation of the mother's diet and lifestyle in order to maximise intrauterine bone mineral accrual and postnatal skeletal growth and thus reduce the burden of osteoporotic fractures in future generations.

Let $\Xi $ be a discrete set in ${{\mathbb{R}}^{d}}$. Call the elements of $\Xi $centers. The well-known Voronoi tessellation partitions ${{\mathbb{R}}^{d}}$ into polyhedral regions (of varying volumes) by allocating each site of ${{\mathbb{R}}^{d}}$ to the closest center. Here we study allocations of ${{\mathbb{R}}^{d}}$ to $\Xi $ in which each center attempts to claim a region of equal volume $\alpha $.

We focus on the case where $\Xi $ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation.

The model exhibits a phase transition in the appetite $\alpha $. In the critical case $\alpha \,=\,1$ we prove a power law upper bound on $X$ in dimension $d\,=\,1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha <1$ and $\alpha \,>1$we prove exponential upper bounds on $X$.