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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.
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{
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 $l
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 othe
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 bo