On asphericity of convex bodies

The Isoperimetric Quotient of a Convex Body Decreases Monotonically Under the Eikonal Abrasion Model

“…We remark that E(K) is a convex body if, and only if there is a unique largest ball inscribed in K, F (K) is contained in a translate of 1 r(K) E(K), and that the quantity t appearing in Theorem 1 has the property that this is the smallest value of t such that this containment is not strict. In the spirit of [11], we will use the following Definition 2. Let K be a convex body and let r(K) and R(K) denote the radius of a largest inscribed and the smallest circumscribed sphere of K, respectively.…”

Uniqueness of centers of nearly spherical bodies

“…It is not necessarily unique, as one can see by considering the example of an isosceles triangle of edge lengths 1, a, a (a > 1), where the set of asphericity centers is the interval on the angle bisector of the smallest angle between two intersection points, one with the bisectors of the longest edges and the other with the angle bisectors of the greatest angles. Dudov and Meshcheryakova [5] showed that the asphericity center is unique if Ω is strictly convex or if Ω is centrally symmetric 1 .…”

Систематизация Задач По Шаровым Оценкам Выпуклого Компакта

“…Известна также задача об асферичности выпуклого тела (см. [14]- [17]). Перспективной для приложений представляется задача о шаровой оболочке наименьшего объема для границы выпуклого тела.…”

“…[34]). В работах [15], [17] получены необходимые и достаточные условия решения задачи (7.1), условия единственности решения, предложен метод приближенного решения. Примеры показывают, что функция ψ(x) может быть и не выпуклой, и не вогнутой даже локально всюду на D.…”