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# 数学代写|凸分析代写Convex Analysis代考|ESE605 Description of the Unified Method

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1. 如果$0 \leq r \leq 1$是实数，并且$x, y\in C$，那么$r x+(1-r) y \in C$。[1]
2. 如果$0<r<1$是实数，并且$x, y\in C$有$x\neq y$，那么$r x+(1-r) y\in C$。

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## 数学代写|凸分析代写Convex Analysis代考|Description of the Unified Method

Description of the Unified Method The foundation of the unified approach is the homogenization method. This method consists of three steps:

1. homogenize, that is, translate a given task into the language of nonnegative homogeneous convex sets, also called convex cones,
2. work with convex cones, which is relatively easy,
3. dehomogenize, that is, translate back to get the answer to the task at hand.
The use of homogenization in convex analysis is borrowed from its use in geometry. Therefore, we first take a look at its use in geometry.

History of the Unified Method As long ago as 200 years BCE, Apollonius of Perga used an embryonic form of homogenization in his eight volume work “Conics,” the apex of ancient Greek mathematics. He showed that totally different curves-circles, ellipses, parabolas, and hyperbolas-have many common properties as they are all conic sections, intersections of a plane with a cone.

Figure 2 shows an ice cream cone and boundaries of intersections with four planes. This gives, for the horizontal plane a circle, for the slightly slanted plane an ellipse, for the plane that is parallel to a ray on the boundary of the cone a parabola, and for the plane that is even more slanted a branch of a hyperbola.

Thus, apparently unrelated curves can be seen to have common properties because they are formed in the same way: as intersections of one cone with various planes. This phenomenon runs parallel to the fact that totally different convex sets-that is, sets in column space $\mathbb{R}^n$ that consist of one piece and have no holes or dents-have many common properties and that this can be explained by homogenization: each convex set is the intersection of a hyperplane and a convex cone, that is, a convex set that is positive homogeneous-containing all positive scalar multiples for each of its points.

## 数学代写|凸分析代写Convex Analysis代考|Working with Unboundedness in Geometry by the Unified Method

Working with Unboundedness in Geometry by the Unified Method In 1415 , the Florentine architect Filippo Brunelleschi made the first picture that used linear perspective. In this technique, horizontal lines that have the same direction intersect at the horizon in one point called the vanishing point.

Figure 4 illustrates linear perspective; it shows a stylized version of parallel tulip fields of different colors that seem to stretch to the horizon. The vanishing point is behind the windmill.

In the early nineteenth century, the technique of linear perspective inspired the discovery of projective geometry. Projective space includes a “horizon” consisting of “points at infinity,” which represent two-sided directions. Projective space enables dealing with problems at infinity in algebraic geometry. This is again an instance of homogenization. Even before the discovery of projective space, Carl Friedrich Gauss made a recommendation about how one should deal with one-sided directions. In the first lines of his “Disquisitiones Generales Circa Superficies Curvas” (general investigations of curved surfaces), published in 1828, the most important work in the history of differential geometry, he wrote:
Disquisitiones, in quibus de directionibus variarum in spazio agitur, plerumque ad maius perspicuitatis et simplicitatis fastigium evehuntur, in auxilium vocando superficiem sphaericum radio $=1$ circa centrum arbitrarium descriptam, cuius singula puncta repraesentare censebuntur directiones rectarum radiis ad illa terminatis parallelarum.

## 数学代写|凸分析代写Convex Analysis代考|Description of the Unified Method

1. 均质化，即将给定任务翻译成非负齐次凸集（也称为凸锥）的语言，
2. 使用凸锥，这相对容易，
3. dehomogenize，即翻译回来得到手头任务的答案。
均匀化在凸分析中的使用是从它在几何中的使用借来的。因此，我们先来看看它在几何中的应用。

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