Hall’s Conjecture on Extremal Sets for Random Triangles

AbstractWe prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

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SETS WITH SEVERAL CENTERS OF SYMMETRY

International Journal of Number Theory ◽

10.1142/s1793042111004174 ◽

2011 ◽

Vol 07 (05) ◽

pp. 1115-1135

Author(s):

GREGORY A. FREIMAN ◽

YONUTZ V. STANCHESCU

Keyword(s):

Finite Subset ◽

Extremal Sets ◽

Finite Set

Let A be a finite subset of the group ℤ2. Let C = {c0, c1,…,cs-1} be a finite set of s distinct points in the plane. For every 0 ≤ i ≤ s -1, we define Di = {a - a′ : a ∈ A, a′ ∈ A, a + a′ = 2ci} and Rs(A) = |D0 ∪ D1 ∪…∪ Ds-1|. In [1, 2], we found the maximal value of Rs(A) in cases s = 1, s = 2 and s = 3 and studied the structure of A assuming that R3(A) is equal or close to its maximal value. In this paper, we examine the case of s = 4 centers of symmetry and we find the maximal value of R4(A). Moreover, in cases when the maximal value is attained, we will describe the structure of extremal sets.

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Existence of extremal Beltrami coefficients with nonconstant modulus

Nagoya Mathematical Journal ◽

10.1215/00277630-2010-001 ◽

2010 ◽

Vol 199 ◽

pp. 1-14 ◽

Cited By ~ 3

Author(s):

Guowu Yao

Keyword(s):

Open Problem ◽

Teichmüller Space ◽

Teichmuller Space ◽

Constant Modulus ◽

Beltrami Coefficient ◽

Extremal Sets ◽

Universal Teichmüller Space

AbstractSuppose that [μ]T(Δ) is a point of the universal Teichmüller space T(Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists μ such that μ is uniquely extremal in [μ]T(Δ) and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in [μ]T(Δ) if [μ]T(Δ) admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich’s construction theorem.

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