Convex To The Origin

The Enigmatic Embrace of Convexity: A Journey to the Origin

Imagine a vast, star-studded expanse. Each star represents a point in a mathematical space. Now, imagine drawing a line connecting any two of these stars. If that line, in its entirety, remains within the constellation of stars, we’re dealing with a fascinating concept: convexity. But what if we add a crucial element – the origin, the central point (0,0) in our mathematical universe? This introduces us to the captivating world of sets that are "convex to the origin," a concept with deeper implications than you might initially suspect.

1. Defining Convexity to the Origin

In simple terms, a set of points is convex to the origin if, for any two points within the set, the line segment connecting them also lies entirely within the set and passes through the origin. Let's break this down:

Convexity: A set is convex if, for any two points within the set, the line segment connecting them is also entirely contained within the set. Think of a circle or an ellipse; no matter which two points you choose, the line connecting them stays within the shape. A star shape, on the other hand, is not convex.

To the Origin: This adds the crucial condition that the line segment must pass through the origin (0,0). This dramatically restricts the types of sets that qualify. A simple circle centred anywhere other than the origin is not convex to the origin.

Consider a simple example: the positive quadrant of a Cartesian plane (where both x and y coordinates are positive). This region is convex to the origin. Pick any two points within this quadrant; the line joining them will always stay within the quadrant and pass through (0,0). However, a circle centered at (1,1) with a radius of 1 is convex but not convex to the origin.

2. Mathematical Representation

Mathematically, a set S is convex to the origin if, for any two points x and y in S, and for any scalar λ between 0 and 1, the point λx + (1-λ)y is also in S. The crucial part is that this condition must hold for all such λ, meaning the entire line segment connecting x and y, including the point where λ = 0 (point y) and λ = 1 (point x), resides within the set and passes through the origin.

This mathematical definition allows for precise analysis and is used extensively in various fields.

3. Applications in Real-World Scenarios

The concept of convexity to the origin, while seemingly abstract, finds practical applications in several domains:

Economics: In production economics, the production possibility frontier (PPF) often exhibits convexity to the origin. This reflects the principle of increasing opportunity cost: as you produce more of one good, you must sacrifice increasingly more of another. The origin signifies no production of either good.

Operations Research: Linear programming, a cornerstone of operations research, heavily relies on the concept of convex sets. Many optimization problems involve finding the optimal solution within a convex region, often related to resource allocation and production planning. Convexity to the origin might represent feasible solutions with resource constraints centred at the origin.

Machine Learning: Convex functions play a vital role in machine learning algorithms. The optimization of many machine learning models involves minimizing a convex cost function. The origin may represent a baseline or a zero-error state.

Image Processing: In image processing, convex hulls are used for shape analysis and object recognition. Convexity to the origin might be relevant when analyzing shapes centered around a specific point of interest.

4. Beyond the Basics: Cones

A special class of sets convex to the origin are cones. A cone is a set that, if it contains a point x, it also contains λx for all λ ≥ 0. In simpler terms, if a point is in the cone, any positive scalar multiple of that point is also in the cone. This means cones always extend infinitely from the origin. Many optimization problems in engineering and economics are formulated using cones, often representing feasible regions.

5. Expanding the Horizon: Non-Convex Sets

It's crucial to understand that not all sets are convex to the origin. Many real-world scenarios involve non-convex sets, making their analysis more complex. However, techniques exist to approximate non-convex sets with convex ones, allowing for easier analysis and optimization.

Reflective Summary

The concept of "convex to the origin" is a fundamental idea in mathematics with surprisingly broad applications across various fields. It's a powerful tool for analyzing sets, understanding constraints, and solving optimization problems. The addition of the "to the origin" condition introduces a crucial constraint, dramatically narrowing the scope of eligible sets and highlighting the importance of the central point in many mathematical and real-world models. From production possibility frontiers in economics to cost functions in machine learning, its relevance spans numerous disciplines. Understanding convexity to the origin empowers us to model and solve complex problems more effectively.

FAQs

1. Q: Can a set be convex but not convex to the origin? A: Yes, absolutely. A circle not centered at the origin is a perfect example.

2. Q: What if the origin is not at (0,0)? A: The definition needs to be adjusted. The origin becomes a specified point, and the line segment connecting any two points in the set must pass through that specific point.

3. Q: Are all cones convex to the origin? A: Yes, all cones are convex to the origin by definition.

4. Q: Why is the concept of "to the origin" important? A: It introduces a critical constraint that significantly influences the type of sets being considered, making the analysis more targeted and relevant to specific problems.

5. Q: How do I determine if a given set is convex to the origin? A: Use the mathematical definition; verify that for any two points in the set, the line segment connecting them remains entirely within the set and passes through the origin. You can also visualize it graphically.

Search Results:

凸（凹）函数、拟凸（凹）函数、伪凸（凹）函数是什么？ - 知乎 Convex function Quasiconvex function Pseudoconvex function 在R上考虑就很容易理解。在凸要求不高过xy连线的地方，拟凸只要求不高过xy中较大者，所以凸蕴含拟凸；而伪凸则要求，“ …

为什么数学概念中，将凸起的函数称为凹函数？ - 知乎 那么我们来讲凸函数（convex function）为什么叫做是凸（convex）的：这是因为凸函数与凸集（convex set）有联系，而凸集的定义没有争议。 1. 凸函数与凸集通过 sublevel sets 这个概念 …

《凸优化》这本书怎么学习或阅读？ - 知乎 Boyd 的《Convex Optimization》确实是一本好书，当年在数学系读书的时候，很多老师也都推荐这本书。这本书的优点是大而全，拿在手上就能感受到沉甸甸的重量。。。我自己也曾经想好 …

如何看待王虹和 Joshua Zahl 在 arXiv 发布对三维挂 … 2025年2月26日，北大校友王虹和Zahl合作，提交了预印本论文Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions，宣告证明了三维的挂谷猜想。

五种凸优化问题取全局最优点判定条件的完整证明推导过程 29 Aug 2023 · 凸优化问题的形式是什么？凸优化问题有什么特征。五种凸优化问题取全局最优点的判定条件，及其推导证明过程。由于看的书是Stephen Boyd的convex optimization，为了 …

凸凹函数到底怎么定义？ - 知乎 24 Jun 2019 · 图像以上为凸集的就是凸函数，如开口向上的二次函数，英文里叫Convex或concave up，反过来就是凹函数，英文叫Concave或concave down，这是国际上的定义。国内 …

为什么 Non-Convex Optimization 受到了越来越大的关注？ - 知乎 为什么 Non-Convex Optimization 受到了越来越大的关注？近几年ICMl,NIPS 出现了许多Non-Convex Optimization的论文，我虽然也在看Non-Convex Optimization… 显示全部关注者 …

凸分析（convex analysis） - 知乎 2.1.1 定义（凸集）空间的一个子集，若对于，，有，则称子集为一个凸的（convex）。根据前述，显然，所有仿射集（包括空集和全空间）都是凸的。

挂谷猜想有多重要？为什么破解它的王虹有望获得菲尔兹奖？ - 知乎 2025年02月24日，纽约大学科朗数学科学研究所助理教授王虹与不列颠哥伦比亚大学助理教授约书亚·扎尔（Joshua Zahl）在预印本网站（arXiv）上，提交了一篇题为《凸集的并集的体积估 …

对Convex Optimization凸优化书上透视函数的一些理解 - 知乎 5 May 2023 · 继续分享自己对Stephen Boyd巨作《Convex Optimization》书上概念的一些理解，由于用的是IPAD看的书，所以推导分析过程就用IPAD屏幕写的，效果不好看请见谅。今天 …