Convex geometry studies the shape of the solution sets arising in combinatorial optimization, analysis, operations research and more. The discipline has a long history, one still being written today thanks to the spectacular advances currently being made in algorithms, optimization, machine learning and artificial intelligence.
With a lasso or an elastic band…
------------------------------
A convex set in the plane or in space has an intuitive geometric property: if two points belong to , then so does every point on the line segment joining them. The notion of a convex hull is just as accessible: if is a set of points in the plane , its convex hull can be viewed as the region of enclosed by a kind of "lasso" wrapped around . If is, for example, the outline of a capital W, the lasso encloses the filled trapezoid containing the letter. Imagine stretching a large elastic band "around" the letter W and letting go: the geometric shape assumed by the band is the convex hull we seek. More formally, the convex hull of a subset of the plane or of space is the intersection—necessarily convex—of all convex sets containing . It is the smallest convex set containing .
For example, in the plane, if is the union of two distinct parallel lines, its convex hull is the region of the plane between those two lines.
Similarly, if is the union in space of three non-coplanar vertical lines—or three non-coplanar parallel lines—its convex hull is the triangular prism whose edges are those lines.
