Polynomials associated to non-convex bodies

Convex Floating Bodies as Approximations of Bergman Sublevel Sets on Tube Domains

For a pseudoconvex tube domain, we prove estimates that relate the sublevel sets of its diagonal Bergman kernel to the floating bodies of its convex base. This allows us to associate a new affine invariant to any convex body.

Extremal functions for real convex bodies

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.

实凸体的极值函数：单纯形、条带与椭圆

Extremal functions for real convex bodies: simplices, strips, and ellipses

本文提出用支撑单纯形和条带显式计算实凸多面体Siciak-Zaharjuta极值函数的方法，并据此给出实凸体极值椭圆存在性的新证明。

Monge-Ampère Measures for Convex Bodies and Bernstein-Markov Type Inequalities

We use geometric methods to calculate a formula for the complex Monge-Ampère measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies $K$. We apply this to show that two methods for deriving Bernstein-Markov-type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area elli…

Bernstein-Walsh theory associated to convex bodies and applications to multivariate approximation theory

We prove a version of the Bernstein-Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full polynomial space associated to a convex body P. As a consequence, we validate and clarify some observations of Trefethen in multivariate approximation theory.

Convex bodies in Euclidean and Weil-Petersson geometries

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting metric structure shares with Thurston's asymmetric metric is noted.