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AbstractWe consider ℂⁿ as a normed space equipped with a complex norm F and we investigate some geometrical problems related with the notion of a conjugate norm F*. A crucial role in our considerations is played by the classical Shmul'yan theorem on exposed points in dual spaces. Many applications of this theorem are given for different problems including characterization of linear (biholomorphic) equivalence for a class of balls in ℂⁿ, calculation of the group of linear automorphisms (Section 4)...

The main result of this paper is the following: if a compact subset E of ${ℝ}^n$ is UPC in the direction of a vector $v\in S^{n-1}$ then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve an earlier result of Pawłucki and Pleśniak [PP1].

We point out relations between Siciak’s homogeneous extremal function ${\Psi }_B$ and the Cauchy-Poisson transform in case $B$ is a ball in ℝ². In particular, we find effective formulas for ${\Psi }_B$ for an important class of balls. These formulas imply that, in general, ${\Psi }_B$ is not a norm in ℂ².

We point out relations between the injective complexification of a real Banach space and polynomial inequalities. In particular we prove a generalization of a classical Szegő inequality to the case of polynomial mappings between Banach spaces. As an application we observe a complex version of known Bernstein-Szegő type inequalities.

We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in ${ℝ}^N$ is Hölder continuous then E admits a Szegö type inequality with weight function $dist{(x,\partial E)}^{-(1-\kappa )}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

It is known that for $C^{\infty }$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty }$ determining. In this paper we give examples of sets which are not $C^{\infty }$ determining, but have the Bernstein and generalized Markov properties.

The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{\nu }\left(E₁\times E₂\right)=min(C_{\nu ₁}\left(E₁\right),C_{\nu ₂}\left(E₂\right))$, where $E_j$ and ${\nu }_j$ are respectively a compact set and a norm in ${ℂ}^{N_j}$ (j = 1,2), and ν is a norm in ${ℂ}^{N₁+N₂}$, ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of ${ℂ}^N$, denote by C(E) the standard L-capacity and by ${\omega }_E$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes in...

Consider the normed space $(\mathbb{P}\left({ℂ}^N\right),|\left|·\right|\left|\right)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_g:(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|\left)\ni f\mapsto gf\in \right(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|\partial /\partial z_j{P\left|\right|\le M\left(degP\right)}^m\left|\right|P\left|\right|$, j = 1,...,N, $P\in \mathbb{P}\left({ℂ}^N\right)$, with positive constants M and m is equivalent to the inequality $\left|\right|{\partial }^N/\partial z₁...\partial z_{N}P\left|\right|\le M^{\text{'}}{\left(degP\right)}^{m^{\text{'}}}\left|\right|P\left|\right|$, $P\in \mathbb{P}\left({ℂ}^N\right)$, with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In...