Approximate extrema of a bounded family of intervals

Устанавливаются связи между экстремальными множествами для ``приближенных'' отношений доминирования для произвольного ограниченного семейства интервалов вещественных чисел. Connections are established between extremal sets for ``approximate'' dominance relations for an arbitrary bounded family of intervals of real numbers.

Two new constructions of approximately mutually unbiased bases

Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.

A survey of a selection of methods for determination of Koebe sets

In this article we take over methods for determination of Koebe set based on extremal sets for a given class of functions.

On characterizations of sausages via inequalities and roots of Steiner polynomials

We prove that sausages are the family of ‘extremal sets’ in relation to certain linear improvements of Minkowski’s first inequality when working with projection/sections assumptions. In particular they characterize the equality cases of the corresponding linear refinements of both the isoperimetric inequality and Urysohn’s inequality. We also characterize sausages by algebraic properties of the roots of Steiner polynomials, in which other functionals of convex bodies such as the inradius, the mean width or the diameter are involved.