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Polynomial spaces associated to a convex body $C$ in $({bf R}^+)^d$ have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex $C$. We develop some basic pluripotential theory including notions of $C-$extremal plurisubharmonic functions $V_{C,K}$ for $Ksubset {bf C}^d$ compact. Using this, we discuss Bernstein-Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.
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We study the smoothness of the Siciak-Zaharjuta extremal function associated to a convex body in $mathbb{R}^2$. We also prove a formula relating the complex equilibrium measure of a convex body in $mathbb{R}^n$ to that of its Robin indicatrix. The main tool we use are extremal ellipses.
For every tuple $d_1,dots, d_lgeq 2,$ let $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}$ denote the tensor product of $mathbb{R}^{d_i},$ $i=1,dots,l.$ Let us denote by $mathcal{B}(d)$ the hyperspace of centrally symmetric convex bodies in $mathbb{R}^d,$ $d=d_1cdots d_l,$ endowed with the Hausdorff distance, and by $mathcal{B}_otimes(d_1,dots,d_l)$ the subset of $mathcal{B}(d)$ consisting of the convex bodies that are closed unit balls of reasonable crossnorms on $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}.$ It is known that $mathcal{B}_otimes(d_1,dots,d_l)$ is a closed, contractible and locally compact subset of $mathcal{B}(d).$ The hyperspace $mathcal{B}_otimes(d_1,dots,d_l)$ is called the space of tensorial bodies. In this work we determine the homeomorphism type of $mathcal{B}_otimes(d_1,dots,d_l).$ We show that even if $mathcal{B}_otimes(d_1,dots,d_l)$ is not convex with respect to the Minkowski sum, it is an Absolute Retract homeomorphic to $mathcal{Q}timesmathbb{R}^p,$ where $mathcal{Q}$ is the Hilbert cube and $p=frac{d_1(d_1+1)+cdots+d_l(d_l+1)}{2}.$ Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to $mathcal{B}_otimes(d_1,dots,d_l)$ is examined.
Integer partitions express the different ways that a positive integer may be written as a sum of other positive integers. Here we explore the analytic properties of a polynomial $f_lambda(x)$ that we call the partition polynomial for the partition $lambda$, with the hope of learning new properties of partitions. We prove a recursive formula for the derivatives of $f_lambda(x)$ involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of $f_lambda(x)$ is dense in $[0,1/2]$, pose a few open questions, and formulate a conjecture relating the integral to the length of the partition. We also provide specific examples throughout to support our speculation that an in-depth analysis of partition polynomials could further strengthen our understanding of partitions.
We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hormander and Radstrom. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.
We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.
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