On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Genetic variants of VDR and CYP2R1 affect BMI independently of serum vitamin D concentrations

BACKGROUND: Vitamin D metabolism and obesity have been linked by several studies, however the reason for this association is unclear. Our objective was to investigate potential correlations between genetic variants in key enzymes of vitamin D metabolism and the body mass index on a representative and random sample of Hungarian adults. METHODS: Altogether 462 severely vitamin D deficient individuals were studied at the end of winter in order to decrease environmental and maximize any relevant genetic effect. Furthermore, participants with lifestyle factors known to affect vitamin D homeostasis were also excluded. We selected 23 target SNPs in five genes that encode key proteins of vitamin D metabolism (NADSYN1, GC, CYP24A1, CYP2R1, VDR). RESULTS: Variants in 2 genetic polymorphisms; rs2853564 (VDR) and rs11023374 (CYP2R1) showed a significant association with participants' BMI. These associations survived further adjustment for total-, free-, or bioactive-25(OH) vitamin D levels, although the variance explained by these 2 SNPS in BMI heterogeneity was only 3.2%. CONCLUSION: Our results show two novel examples of the relationship between genetics of vitamin D and BMI, highlighting the potential role of vitamin D hormone in the physiology of obesity

Why not to use the handgrip test in the assessment of cardiovascular autonomic neuropathy among patients with diabetes mellitus?

OBJECTIVE: Historically, a set of 5 cardiovascular autonomic reflex tests (CARTs) was considered to be the gold standard in the assessment of cardiovascular autonomic neuropathy (CAN). However, measuring diastolic blood pressure (BP) response to sustained handgrip is omitted in recent guidelines. We aimed to assess the association between the handgrip and the other 4 tests as well as to identify determinants of the handgrip test results in diabetic patients. PATIENTS AND METHODS: 353 patients with diabetes (DM) were recruited (age: 60.2±7.4 years; female: 57.2%; BMI: 29.3±2.1 kg/m2; DM duration: 15.6±9.9 years; HbA1c: 7.8±1.4% (66 mmol/mol); with type 1 DM: 18.1%). CAN was assessed by 5 CARTs: the deep breathing test, Valsalva ratio, 30/15 ratio, handgrip and orthostatic hypotension test. RESULTS: Sensitivity and specificity of the handgrip test in the diagnosis of definite CAN were 24.6% (95%CI 17.7-33.1%) and 79.4% (95%CI 73.3-84.4%), respectively. Results of the handgrip test did not show any association with those of the deep-breathing test (γ=0.004, p=0.563), 30/15 ratio (γ=0.282, p=0.357), Valsalva ratio (γ=-0.058, p=0.436) and orthostatic hypotension (γ=-0.026, p=0.833). Handgrip test abnormality showed an independent association with higher initial diastolic BP (OR 1.05, p=0.0009) and an independent inverse association with the presence of hypertension (OR=0.42, p=0.006). CONCLUSIONS: Our data confirm that the handgrip test should no longer be part of the cardiovascular autonomic testing being highly dependent on hypertensive status and baseline diastolic BP. Exaggerated exercise pressor response is proposed as putative mechanism for the inverse association between abnormal results of the handgrip test and hypertension. Adequate CARTs important to allow their use in clinical trials and for the prevention of DM-associated complications by initiating early treatment

A characterization of affinely regular polygons

In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $\lambda \geq 0$ such that $p_{j+2}-p_{j-1} = \lambda (p_{j+1}-p_j)$ for $j=1,2,\ldots, n$. The aim of this paper is to examine the properties of polygons whose vertices $p_1,p_2,\ldots,p_n \in \mathbb{C}$ satisfy the property that $p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)$ for some $w \in \mathbb{C}$ and $m_1,m_2,k \in \mathbb{Z}$. In particular, we show that in `most' cases this implies that the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups.Comment: 11 pages, 1 figur