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# 数学代写|凸分析代写Convex Analysis代考|CS675 Convex Hull and Conic Hull

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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|Convex Hull and Conic Hull

Here is a source of many basic examples of convex sets.
Definition 1.7.2 To each set $S \subseteq X=\mathbb{R}^n$, one can add points from $X$ in a minimal way in order to make a convex set $\operatorname{co}(S) \subseteq X$, the convex hull of $S$. That is, $\operatorname{co}(S)$ is the smallest convex set containing $S$. It is the intersection of all convex sets containing $S$.
Example 1.7.3 Figure $1.15$ illustrates the concept of convex hull.
From left to right: the first two sets are already convex, so they are equal to their convex hull; for the other convex sets, taking their convex hull means respectively filling a dent, filling a hole, and making from a set consisting of two pieces a set consisting of one piece, without dents.
Here is a similar source of examples of convex cones.
Definition 1.7.4 To each set $S \subseteq X$, one can adjoin to $S \cup\left{0_X\right}$ points from $X$ in a minimal way in order to make a convex cone cone $(S)$, the conic hull of $S$. That is, cone $(S)$ is the smallest convex cone containing $S$ and the origin. It is the intersection of all convex cones containing $S$ and the origin.
Example 1.7.5 (Conic Hull)

1. Figure $1.16$ illustrates the concept of conic hull.
This picture makes clear that a lot of structure can get lost if we pass from a set to its conic hull.
2. The conification of a convex set $A \subset \mathbb{R}^n$ is essentially the conic hull of $A \times{1}$ :
$$c(A)=\operatorname{cone}(A \times{1}) \backslash\left{0_{X \times \mathbb{R}}\right}$$

## 数学代写|凸分析代写Convex Analysis代考|Sphere Model for a Convex Cone

One often chooses a representative for each open ray, in order to replace open rayswhich are infinite sets of points-by single points. A convenient way to do this is to normalize: that is, to choose the unit vector on each ray. Thus the one-sided directions in the space $X=\mathbb{R}^n$ are modeled as the points on the standard unit sphere $S_X=S_n={x \in X \mid|x|=1}$ in $X$. The set of unit vectors in a convex cone $C \subseteq X=\mathbb{R}^n$ will be called the sphere model for $C$. The subsets of the standard unit sphere $S_X$ that one gets in this way are precisely the geodesically convex subsets of $S_X$. A subset $T$ of $S_X$ is called geodesically convex if for each two different points $p, q$ of $T$ that are not antipodes $(p \neq-q)$, the shortest curve on $S_X$ that connects them is entirely contained in $T$. This curve is called the geodesic connecting these two points. Note that for two different points $p, q$ on $S_X$ that are not antipodes, there is a unique great circle on $S_X$ that contains them. A great circle on $S_X$ is a circle on $S_X$ with center the origin, that is, it is the intersection of the sphere $S_X$ with a two dimensional subspace of $X$. This great circle through $p$ and $q$ gives two curves on $S_X$ on this circle connecting the two points, a short one and a long one. The short one is the geodesic connecting these points.
Example 1.4.3 (Sphere Model for a Convex Cone)

1. Figure $1.5$ above illustrates the sphere model for convex cones in dimension two. The convex cone $C$ is modeled by an arc.
2. Figure $1.6$ illustrates the sphere model for convex cones in dimension three, and it illustrates the concepts great circle and geodesic on $S_X$ for $X=\mathbb{R}^3$.

Two great circles are drawn. These model two convex cones that are planes through the origin. For the two marked points on one of these circles, the segment on this circle on the front of the sphere connecting the two points is their geodesic. Moreover, you see that two planes in $\mathbb{R}^3$ through the origin (and remember that a plane is a convex cone) are modeled in the sphere model for convex cones by two large circles on the sphere.

1. Figure $1.7$ is another illustration of the sphere model for a convex cone $C$ in dimension two.

Here the convex cone $C$ is again modeled by an arc. The sphere model is convenient for visualizing the one-sided directions determined by a convex cone $C \subseteq X=\mathbb{R}^n$ for dimension $n$ up to three: for $n=3$, a one-sided direction in

## 数学代写|凸分析代写Convex Analysis代考|Convex Hull and Conic Hull

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