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数学代写|复分析代写Complex analysis代考|Why is mathematical morphology a useful tool in the study of convexity?

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数学代写|复分析代写Complex analysis代考|Why is mathematical morphology a useful tool in the study of convexity?

Mathematical morphology can be superficially described as applied lattice theory. As such it is about the operations $(x, y) \mapsto x \wedge y=\min (x, y)$ and $(x, y) \mapsto x \vee y=\max (x, y)$ in an ordered set. These operations replace addition and multiplication in a ring, and an important example is the Boolean ring of all subsets of a given set, with $\wedge$ as intersection and $\vee$ as union.

In convexity theory, we see that the intersection of two convex sets is convex, while the union is, in general, not. But we still have a lattice, in that the convex hull of the union of two convex sets is the smallest convex set containing the two, and therefore is the supremum of the two. This can then be done analogously for convex functions, and, more generally for plurisubharmonic functions. It turns out that complete lattices are important; they are the ordered sets which allow infima and suprema also of infinite families.

Mathematical morphology provides us with important concepts in the theory of ordered sets that are helpful in understanding several related phenomena in mathematics. See Section 9.3 for more details.

数学代写|复分析代写Complex analysis代考|Which are the most significant results reported in the present chapter?

An important observation is the non-local character of lineal convexity for general sets. As always, properties such that the local and global variants are different create difficulties-which may be challenging.

Because of the non-local character just mentioned, it is of importance to know that for bounded sets with a smooth boundary, the property of being locally lineally convex actually implies the global property. This is proved in Section 9.6.

What makes the approach in the present chapter different from other presentations of the subject?

Complex convexity is quite a well-studied field, and the ways to approach it are not many. However, we shall view convexity from the inside as well as from the outside, and this gives perhaps interesting perspectives. A set is concave if and only if its complement is convex, and the two notions should be studied together.

As mentioned, a subset $A$ of $\mathbf{R}^n$ is defined to be convex if for any pair ${a, b}$ of points in $A$, the whole segment $[a, b]$ is also contained in $A$. This is what we can call convexity from the inside, i.e., looking at subsets of the given set. But we can also look at the set from the outside: If $p$ does not belong to $A$ and $A$ is open or closed, then there is a half-space that contains $A$ but not $p$. This is the Hahn-Banach theorem, of utmost important in convexity theory, both in finite dimension and infinite dimension. Explicitly, we say that a set in a vector space over $\mathbf{R}$ or $\mathbf{C}$ is lineally concave if it is a union of hyperplanes, and lineally convex if its complement is lineally concave. In one dimension, hyperplanes are just points, so every set is both lineally concave and lineally convex. In higher dimensions a convex set need not be lineally convex, but if it is open or closed, this is true. All this is true both in the real and the complex settings. For more details, see Section 9.4.

数学代写|复分析代写Complex analysis代考|Why is mathematical morphology a useful tool in the study of convexity?

$(x, y) \mapsto x \vee y=\max (x, y)$ 在一个有序的集合中。这些操作取代了环中的加法和乘法，一个重要的例子是 给定集合的所有子集的布尔环， $\wedge$ 作为交集和 $\vee$ 作为联盟。

数学代写|复分析代写Complex analysis代考|Which are the most significant results reported in the present chapter?

$-一$ 这可能具有挑战性。

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