ashina14 said:Homework Statement
Prove that F = {x E R^n : Ax >/= b; x >/= 0} is a convex
set.
Yes x in non negative and A and b are any arbitrary
ashina14 said:Homework Statement
Prove that F = {x E R^n : Ax >/= b; x >/= 0} is a convex
set.
LCKurtz said:Are arbitrary what? What does it mean for ##x\in R^n## to satisfy ##x > 0##?
ashina14 said:A and b are any arbitrary number in R.
All vectors in x E Rn are non-negative
A convex set is a set of points where, for any two points within the set, the line segment connecting them is also within the set. In other words, all points on the line connecting any two points in the set are also part of the set.
To prove a set is convex, you must show that for any two points within the set, the line segment connecting them is also within the set. This can be done by using the definition of convexity and showing that it holds true for all points in the set.
Convex sets are important in science because they have many real-world applications, such as in optimization problems, economics, and physics. They also have useful properties, such as having a unique minimum and being easy to analyze mathematically.
No, a set cannot be both convex and concave. A convex set is one where all points on the line segment connecting any two points in the set are also part of the set, while a concave set is one where all points on the line segment connecting any two points in the set are not part of the set. These are two opposite properties and cannot both be true for the same set.
Convex sets can be used in a variety of ways in research or experiments. For example, they can be used to model optimization problems, analyze economic systems, or understand the behavior of physical systems. By understanding and utilizing convex sets, you can solve complex problems and gain insights into your research or experiments.