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Concave and Convex Functions
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
The set xy¯ $ \overline{xy} $ is the line segment joining the points x and y in X. Then we have the following definition: The set Y⊂X $ Y\subset X $ is said to be a convex set if Y contains the line segment xy¯ $ \overline{xy} $ whenever x and y are two arbitrary points in Y (see Figure 3.1). A convex set is called a convex body if it contains at least one interior point, i.e., if it completely contains some sphere.
Class Groups
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
Remark 3.12 Some background to the language used above is in order. The term convex body refers to a nonempty, convex bounded and closed subset S of Rn. The topological term “closed” means that every accumulation point of a sequence of elements in S must also be in S. This is tantamount to saying that S is closed in the topological space Rn, with its natural topology. However, we do not need to concern ourselves here with this, since it is possible to state and prove the result without such topological considerations. It can also be shown that if S is “compact,” namely every “cover” (a union of sets containing S) contains a finite cover, then it suffices to assume that V(S)≥2nV(P).
The minimal Orlicz mean width of convex bodies
Published in Applicable Analysis, 2022
Let us discuss one more problem of the same nature. Let K be a symmetric convex body in and be the corresponding norm. Assume that for every , where a and b are the smallest positive numbers for which this inequality holds true for every . It is clear that , and we are interested in The condition means that but there exist contact points of TK and B.
Some notes on directional curvature of a convex body in ℝ n
Published in Optimization, 2022
For a compact convex set with in its interior (briefly called convex body), we denote by and the interior of F and the boundary of F, respectively. By , we denote the polar set of F, i.e., and by , the Minkowski functional of F, It is easy to see that that is, the Minkowski functional of F coincides with the support function of . Consequently, where . Using (1) and (2), it is easy to prove the Lipschitz property, In what follows we also use the so-called duality mapping that associates to each the set of all functionals that support at , Denoting by , the normal cone to F at ξ in the sense of convex analysis, it is easy to show that is the pre-image of calculated at ξ.
Novel trend of mixed Minkowski volumes applications
Published in Applicable Analysis, 2020
A. A. Martynyuk, I. M. Stamova
The convex body is uniquely defined by a support function , where denotes the standard scalar product in and is the unit sphere in .