Convex Sets and Convex Functions A convex set is a set of points for which the distance from any point in the set to the set's boundary is always greater...
A convex set is a set of points for which the distance from any point in the set to the set's boundary is always greater than or equal to the distance from that point to the closest point on the boundary. In other words, the distance to the boundary is larger than or equal to the distance to the closest point on the boundary.
Examples of convex sets:
The set of all points in the first quadrant (excluding the origin) is a convex set.
The set of all points equidistant from the points (1, 1) and (3, 3) is a convex set.
The set of all points with coordinates (x, y) such that x^2 + y^2 ≤ 1 is a convex set.
A convex function is a function whose graph is a convex set. In other words, the level sets of a convex function are convex sets.
Examples of convex functions:
The function f(x, y) = x^2 + y^2 is a convex function.
The function f(x) = x^3 - 3x + 1 is a convex function.
The function f(x) = |x| is a non-convex function.
The connection between convex sets and convex functions is important in optimization problems. In optimization problems, we often seek the point in the set that is closest to a given point, or the point that maximizes or minimizes a given function. By using tools from linear algebra, we can determine whether a point is in a convex set and, if so, find the point that is closest to the given point