a convex body

Picture a lump of clay with no dents, no holes, and no spikes: wherever you pick two points inside it, the straight segment joining them never pokes outside. That no-dents property is convexity, and a convex body is the cleanest case of it — a convex set in R^n that is also bounded (it fits inside some big ball), closed (it contains its own boundary skin), and fat enough to have a genuine interior (it is not flattened into a lower-dimensional slab). Balls, cubes, ellipsoids, and solid simplices are convex bodies; a doughnut, a star, or a flat disk sitting inside R^3 are not.

Formally a set K in R^n is convex if for all x, y in K and all t in [0, 1] the point t*x + (1 - t)*y lies in K. K is a convex body if it is convex, compact, and has nonempty interior. Such a body is pinned down by its support function h_K(u) = sup over x in K of the inner product <u, x>, which records, for each unit direction u, how far K reaches in that direction; h_K is convex and positively homogeneous, and it determines K completely. The boundary need not be smooth (a cube has edges and corners), but at every boundary point there is at least one supporting hyperplane touching K and leaving all of K on one side.

Convex bodies are the central objects of convex geometry because so many quantities behave tamely on them: volume, surface area, width, and inradius are all monotone under inclusion, and operations like Minkowski sum (adding bodies pointwise) keep you inside the class. They are also the right setting for the deep volume inequalities — Brunn-Minkowski, the isoperimetric inequality, mixed volumes — and for the geometry of numbers. One honest caveat: convexity is a strong, almost rigid assumption; most everyday shapes are non-convex, and the theory's elegance comes precisely from refusing them, so 'reduce to the convex case' is often a real loss of generality, not a free simplification.