a supporting hyperplane
Rest a flat ruler against the outside of a circle. It touches at exactly one point, and the whole circle lies on one side of the ruler — the ruler never cuts into the disk. Lean a flat board against any convex blob and the same thing happens: it kisses the boundary while keeping the entire blob on one side. That touching flat is a supporting hyperplane. In 2D it is a supporting LINE, in 3D a supporting PLANE, and in higher dimensions the general word is hyperplane.
Precisely: a hyperplane H supports a convex set C at a boundary point P if P lies on H AND all of C lies in one of the two closed half-spaces bounded by H. In other words H touches C (at P, and maybe along a whole flat face) but never slices through its interior; C sits entirely on one side. The central theorem — the supporting hyperplane theorem — guarantees that through EVERY boundary point of a convex set there is at least one supporting hyperplane. At a smooth boundary point it is unique (the tangent), but at a corner, like the vertex of a square, there are infinitely many supporting lines fanning through that corner, which is perfectly allowed. Closely related is the separating hyperplane theorem: two disjoint convex sets can always be cleanly split by a hyperplane with one set on each side.
This is the geometric engine behind an astonishing range of results. In optimization, the supporting hyperplane at an optimum is what 'prices' the constraints (the language of Lagrange multipliers and duality); in machine learning, a separating hyperplane is exactly what a support vector machine searches for to divide two classes of data; in economics it underlies the second welfare theorem. The honest caveat: supporting hyperplanes are a feature of CONVEX sets specifically. At a dent of a non-convex shape, no such touching-from-outside flat exists, because any flat through a concave boundary point necessarily cuts into the region — which is one more reason convexity is so prized.
Take the unit disk x^2 + y^2 <= 1 and its boundary point (1, 0). The vertical line x = 1 supports it: the point (1, 0) lies on the line, and every point of the disk has x <= 1, so the whole disk sits on one side. At the smooth boundary this supporting line is unique. But take the square corner (1, 1) of the unit square: lines of many different slopes all pass through that corner with the square on one side, so the supporting line there is far from unique.
Smooth boundary points have one supporting hyperplane; corners have a whole fan of them.
Supporting hyperplanes exist at every boundary point only for convex sets. Try it on a non-convex shape's dent and you will fail — any flat through a concave point cuts into the set, which is precisely why these theorems require convexity.