I know I need to show the intersection of the two half planes is convex but I do not know how to do this.
A convex set is a set in which any line segment connecting two points within the set lies completely within the set itself. In other words, if you were to draw a line between any two points within a convex set, all points along that line would also be within the set.
To prove a set is convex, you must show that for any two points within the set, all points along the line connecting them are also within the set. This can be done by using the definition of a convex set and showing that it holds true for all possible points within the set.
Proving that a set is convex is important in many areas of mathematics and science. It allows us to make assumptions and inferences about the properties of the set, and can help in solving optimization and decision-making problems.
No, a set cannot be both convex and non-convex. A set is either convex or non-convex based on the definition of a convex set. If a set does not meet the criteria of a convex set, then it is considered non-convex.
The convexity of a set is directly related to its shape. A convex set has a smoother, more curved shape compared to a non-convex set which may have sharp angles or protrusions. The shape of a convex set is also symmetrical, meaning it looks the same from all angles.