a convex set

Stretch a rubber band around a shape and look at whether the shape has any dents. A disk, a filled triangle, a solid ball, a slab — none of these has a dent: pick any two points inside and the straight segment between them stays entirely inside. A crescent moon or a star shape DOES have dents: a segment between two of its points can poke outside. The dent-free shapes are exactly the convex sets, and this single property is one of the most useful in all of applied mathematics.

The definition is disarmingly short. A set C is convex if, for every pair of points P and Q in C, the entire line segment PQ lies in C. Equivalently, every point of the form (1 - t) times P plus t times Q, for t running from 0 to 1, must belong to C; that expression simply sweeps out the segment from P (at t = 0) to Q (at t = 1). That is the whole definition — no calculus, no smoothness needed. It applies in the plane, in space, and in any number of dimensions identically. Lines, half-planes, disks, balls, cubes, and the solutions of any system of linear inequalities are all convex; the intersection of convex sets is again convex (an enormously handy fact), though their union usually is not.

Convexity matters because it is the condition that makes optimization tractable: when you minimize a convex function over a convex set, any local minimum is automatically the global minimum, so you can trust a downhill search to find the true best answer. This is the backbone of modern optimization, economics, machine learning, and operations research. A common slip is to confuse 'convex' with 'curved outward like a lens'. A square is convex despite being all flat sides and sharp corners; convexity is about no dents, not about being rounded — and a perfectly smooth crescent is NOT convex despite its graceful curves.