Product property of global $P$-extremal functions

In this note, we establish a product property for $P$-extremal functions in the same spirit as the original product formula due to J. Siciak in Ann. Polon. Math., 39 (1981), 175–211. As a consequence, we obtain convexity for the sublevel sets of such extremal functions. Moreover, we also generalize the product property of $P$-extremal functions established by L. Bos and N. Levenberg in Comput. Methods Funct. Theory 18 (2018), 361–388, and later by N. Levenberg and M. Perera, in Contemporary Mathematics 743 (2020), 11–19, in which no restriction on $P$ is needed.

A step in the Delaunay mosaic of order k

Given a locally finite set $$X \subseteq {{\mathbb {R}}}^d$$ X ⊆ R d and an integer $$k \ge 0$$ k ≥ 0 , we consider the function $${\mathbf{w}_{k}^{}} :{\mathrm{Del}_{k}{({X})}} \rightarrow {{\mathbb {R}}}$$ w k : Del k ( X ) → R on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.

Nonnegative forms with sublevel sets of minimal volume

We show that the Euclidean ball has the smallest volume among sublevel sets of nonnegative forms of bounded Bombieri norm as well as among sublevel sets of sum of squares forms whose Gram matrix has bounded Frobenius or nuclear (or, more generally, p-Schatten) norm. These volume-minimizing properties of the Euclidean ball with respect to its representation (as a sublevel set of a form of fixed even degree) complement its numerous intrinsic geometric properties. We also provide a probabilistic interpretation of the results.