a supporting hyperplane
Lean a flat board against a smooth boulder so it just touches without cutting in. The board is a plane that meets the boulder at a boundary point and keeps the whole boulder on one side of it. That is a supporting hyperplane: a flat wall propped exactly against a convex body, touching its boundary but never slicing through its interior. For a ball, the supporting hyperplane at a boundary point is the usual tangent plane; for a cube, a single corner has a whole fan of supporting hyperplanes, because the boundary there is not smooth.
Precisely, let K be a convex set in R^n and p a point of its boundary. A hyperplane H = {x : <u, x> = c} (with u a nonzero normal and c a constant) supports K at p if p lies in H and K lies entirely in one of the two closed half-spaces H bounds, say <u, x> <= c for all x in K, with equality at p. Equivalently, c = h_K(u) is the support function value and p is a point where that supremum is attained. The supporting hyperplane theorem guarantees that through EVERY boundary point of a convex body at least one such hyperplane passes — convexity is exactly the condition that makes this always possible.
Supporting hyperplanes are the local face of the separation principle and the geometric heart of duality and optimization. They turn a convex body into the intersection of all its supporting half-spaces, which is the dual description matching the support function. In optimization they are the supporting hyperplanes of the feasible set at an optimum that certify a solution; in functional analysis the same idea, pushed to infinite dimensions, is the geometric Hahn-Banach theorem. A common misconception: the supporting hyperplane at a point need not be unique — it is unique exactly at boundary points where K is 'smooth' (has a single outward normal); at edges and vertices many supporting hyperplanes coexist, fanning over the normal cone.
For the unit square K = [0,1] x [0,1] in R^2, the line y = 1 supports K at every point of the top edge: K lies in the half-plane y <= 1 and touches the line along that whole edge. At the corner (1,1) there is no single tangent line — instead every line through (1,1) with outward normal pointing into the upper-right quadrant supports K, a whole one-parameter family of supporting lines.
An edge has one supporting line; a corner has a whole fan of them.
A supporting hyperplane must leave ALL of K on one side, not merely touch it: a secant plane through the interior is not supporting even though it meets the boundary. And supporting hyperplanes exist at every boundary point only for convex sets — non-convex bodies generally have boundary points no flat can support.