相关论文: Polynomial approximation on convex subsets of $\ma…

The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\partial K$. Let ${\cal A}(K)$ denote the space of all functions $f$ on $K$…

Let \phi be a real-valued valuation on the family of compact convex subsets of \mathbb{R}^n and let K be a convex body in \mathbb{R}^n. We introduce the \phi -covariogram g_{K,\phi} of K as the function associating to each x \in…

Let $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of…

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where…

We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold K in C^n is of polynomial growth (e.g, in dimension one, K is a finite union of transverse Jordan…

This paper provides a discrete Poincar\'e inequality in $n$ space dimensions on a simplex $K$ with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on $K$ and all integrals…

Given Banach spaces E and F, we denote by ${\mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on…

For each positive integer $m$ and each real finite dimensional Banach space $X$, we set $\beta(X,m)$ to be the infimum of $\delta\in (0,1]$ such that each set $A\subset X$ having diameter $1$ can be represented as the union of $m$ subsets…

The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…

Let $M$ be a compact Riemannian manifold with boundary. We show that $M$ is Gromov-Hausdorff close to a convex Euclidean region $D$ of the same dimension if the boundary distance function of $M$ is $C^1$-close to that of $D$. More…

Two of the authors have defined the class $ WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and…

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…

Using the KKM technique, we establish some existence results for variational-hemivariational inequalities involving monotone set valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive several…

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…

Let $K/\mathbb{Q}$ be a finite extension. We prove that the minimal height of polynomials of degree $n$ of which all roots are in $K^\times$ increases exponentially in $n$. We determine the implied constant exactly for totally real $K$ and…

For a Minkowski centered convex compact set $K$ we define $\alpha(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of…

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in…

For a convex body $K\subset\mathbb{R}^d$ the mean distance $\Delta(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for…

We study the approximation of holomorphic functions of several complex variables by the ring $\mathcal{P}^S(\mathbb{C}^n)$ of polynomials whose exponents are restricted to a convex cone $\mathbb{R}_+S$ for some compact convex $S\in…

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…