Width asymptotics for a pair of Reinhardt domains

Aytuna, Aydın; Rashkovskii, Alexander; Zahariuta, V.

doi:10.4064/ap78-1-4

Cited by 10 publications

(12 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for some J with #J > n. Then, by Lemma 3, (1) i | α for some z (0) and z (1) such that u(z (0) ) = 0 and u(z (1) ) = 1. Hence, by what is proved above, there is j 0 ∈ J such that…”

Section: And the Function γ(θ ζ) Is Continuous In θ On The Compact Smentioning

confidence: 99%

“…and G is relatively compact in U. The quadruple [U, f ; r (0) , r (1) ] is called the frame of the pair (L, G); to stress that the polyhedral pair is generated by m analytic functions we will speak about m-polyhedral pairs or m-frames.…”

Section: ω(Z) = ω(D K; Z) = Limmentioning

confidence: 99%

“…For the one-dimensional case, this conjecture is equivalent to Kolmogorov's conjecture about the asymptotics of the ε-entropy of the compact set A D K ; see, e.g., [24] for the history of the problem in that case. The conjecture was proved in [1] for Reinhardt pairs (K, D), using the Rauch-Taylor result about the computation of the real Monge-Ampère operator from convex functions.…”

Section: Proof Of Theorem 4 We Shall Use the Notationmentioning

confidence: 99%

“…ζ→z sup{u(ζ) : u ∈ P (D), u ≤ 1, u| K ≤ 0}, (1) where P (D) stands for the set of all plurisubharmonic functions in D.…”

Section: ω(Z) = ω(D K; Z) = Limmentioning

confidence: 99%

“…Reduction of analytic polyhedral pairs. Any similar analytic m-polyhedral pair with a frame [D, f ; r (0) , r (1) ] can be represented by its normalized frame:…”

Section: ω(Z) = ω(D K; Z) = Limmentioning

confidence: 99%

See 4 more Smart Citations

On approximation by special analytic polyhedral pairs

Zahariuta

2003

Ann. Polon. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. For bounded logarithmically convex Reinhardt pairs "compact set -domain" (K, D) we solve positively the problem on simultaneous approximation of such a pair by a pair of special analytic polyhedra, generated by the same polynomial mapping f : D → C n , n = dim Ω. This problem is closely connected with the problem of approximation of the pluripotential ω(D, K; z) by pluripotentials with a finite set of isolated logarithmic singularities ([23, 24]). The latter problem has been solved recently for arbitrary pluriregular pairs "compact set -domain" (K, D) by Poletsky [12] and S. Nivoche [10,11], while the first one is still open in the general case.

show abstract

“…for some J with #J > n. Then, by Lemma 3, (1) i | α for some z (0) and z (1) such that u(z (0) ) = 0 and u(z (1) ) = 1. Hence, by what is proved above, there is j 0 ∈ J such that…”

Section: And the Function γ(θ ζ) Is Continuous In θ On The Compact Smentioning

confidence: 99%

Section: ω(Z) = ω(D K; Z) = Limmentioning

confidence: 99%

Section: Proof Of Theorem 4 We Shall Use the Notationmentioning

confidence: 99%

“…ζ→z sup{u(ζ) : u ∈ P (D), u ≤ 1, u| K ≤ 0}, (1) where P (D) stands for the set of all plurisubharmonic functions in D.…”

Section: ω(Z) = ω(D K; Z) = Limmentioning

confidence: 99%

“…Reduction of analytic polyhedral pairs. Any similar analytic m-polyhedral pair with a frame [D, f ; r (0) , r (1) ] can be represented by its normalized frame:…”

Section: ω(Z) = ω(D K; Z) = Limmentioning

confidence: 99%

See 3 more Smart Citations

On approximation by special analytic polyhedral pairs

Zahariuta

2003

Ann. Polon. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Plurisubharmonic geodesics and interpolating sets

Cordero–Erausquin

Rashkovskii

2019

Arch. Math.

View full text Add to dashboard Cite

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of C n . Namely, two non-pluripolar, polynomially closed, compact subsets of C n are interpolated as level sets L t = {z : u t (z) = −1} for the geodesic u t between their relative extremal functions with respect to any ambient bounded domain. The sets L t are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge-Ampère capacities of L t satisfy a dual Brunn-Minkowski inequality.

show abstract

Interpolation of Weighted Extremal Functions

Rashkovskii

2021

Arnold Math J.

View full text Add to dashboard Cite

An approach to interpolation of compact subsets of $${{\mathbb {C}}}^n$$ C n , including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities.

show abstract

Width asymptotics for a pair of Reinhardt domains

Abstract: Abstract. For complete Reinhardt pairs "compact set -domain" K ⊂ D in C n , we prove Zahariuta's conjecture about the exact asymptotics

Cited by 10 publications

References 11 publications

On approximation by special analytic polyhedral pairs

On approximation by special analytic polyhedral pairs

Plurisubharmonic geodesics and interpolating sets

Interpolation of Weighted Extremal Functions

Contact Info

Product

Resources

About