# convex sets

(7 hours to learn)

## Summary

A set S in R^d is convex if for any two points x and y in S, the line segment connecting x and y is also contained in S. Convex sets are part of the definition of convex optimization problems, a very general class of optimization problems for which the optimal solution can often be found.

## Context

This concept has the prerequisites:

- vectors (Convex sets are defined in terms of vectors.)
- dot product (The Euclidean ball is a canonical example of a convex set.)

## Goals

- Know how convex sets are defined, both
- geometrically (line segments lie within the set)
- algebraically (in terms of linear combinations of vectors)

- Know some common examples of convex sets (e.g. hyperplanes, halfspaces, Euclidean balls)

## Core resources (read/watch one of the following)

## -Free-

→ Convex Optimization

A graduate-level textbook on convex optimization.

Other notes:

- Some of the examples may involve math you haven't seen; if so, just try to get the gist.

## See also

- Another important notion of convexity is convex functions .
- Convex sets are used to define convex optimization problems , a very general class of optimization problem.
- Some important examples of convex sets:
- solution sets of linear systems
- polytopes
- positive semidefinite cone , a central object in semidefinite programming