The main facts about Hausdorff and packing measures and dimensions of a Borel set$E$are revisited, using$determining$set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$ , where $\mathcal{B}_E$ is the family of all balls centred on$E$and α is a real parameter. With mild assumptions on φ_{α}, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set$V$in ℝ^{$D$}, these notions are used to introduce the$interior$and$exterior$measures and dimensions of any Borel subset of ∂$V$. We stress that these dimensions depend on the choice of φ_{α}. Two determining functions are considered, φ_{α}($B$)=Vol_{$D$}($B$∩$V$)diam($B$)^{α-$D$}and φ_{α}($B$)=Vol_{$D$}($B$∩$V$)^{α/$D$}, where Vol_{$D$}denotes the$D$-dimensional volume.

We construct a 2-dimensional complex manifold$X$which is the increasing union of proper subdomains that are biholomorphic to ℂ^{2}, but$X$is not Stein.

We study the tropical lines contained in smooth tropical surfaces in ℝ^{3}. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.

We prove that spherical spectral analysis and synthesis hold in Damek–Ricci spaces and derive two-radius theorems.

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂ^{$N$}. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter,$Math. Ann.$$337$(2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂ^{$N$}. Our results extend to certain weights and measures defined on cones in ℝ^{$N$}.