a polytope
A polytope is the higher-dimensional cousin of a polygon and a polyhedron: a shape with flat sides and sharp corners, no curves anywhere. A triangle and a square are 2-dimensional polytopes; a cube, a tetrahedron, and a pyramid are 3-dimensional ones; in four dimensions and beyond the family continues with objects like the 4-cube and the 600-cell. The unifying idea is 'a bounded shape carved out by finitely many flat cuts, whose boundary is assembled from flat faces of every dimension below the top.'
There are two equivalent definitions, and their equivalence (the Minkowski-Weyl theorem) is a small miracle. The V-description: a polytope is the convex hull of finitely many points, conv{v_1, ..., v_k} — build it from vertices. The H-description: a polytope is a bounded intersection of finitely many closed half-spaces, {x : <a_i, x> <= b_i, i = 1..m} — carve it with hyperplane cuts. Each cut that actually touches the boundary contributes a face; the faces of a polytope (vertices of dimension 0, edges of dimension 1, ..., facets of dimension n-1) are themselves polytopes, and they organize into the face lattice ordered by inclusion. A cube, for instance, has 8 vertices, 12 edges, 6 facets, plus the empty face and the whole cube.
Polytopes are the meeting point of geometry, combinatorics, and optimization. Linear programming lives on them: the feasible region of a linear program is a polytope (or unbounded polyhedron), and the optimum is attained at a vertex, which is what the simplex method walks toward. Their combinatorics is deep — Euler's relation V - E + F = 2 for 3-polytopes, the Dehn-Sommerville relations and the upper-bound theorem in general. Two cautions: 'polyhedron' is often used for the possibly-unbounded version (a polytope is exactly a bounded polyhedron), and not every combinatorial face-pattern is realizable by an actual geometric polytope, especially in high dimensions — the realizability question is genuinely hard.
The standard simplex in R^3 is conv{(0,0,0), (1,0,0), (0,1,0), (0,0,1)} (V-description): a tetrahedron with 4 vertices. The same solid is also {x in R^3 : x_1 >= 0, x_2 >= 0, x_3 >= 0, x_1 + x_2 + x_3 <= 1} (H-description): four half-space cuts. The two pictures describe one polytope — the Minkowski-Weyl bridge between 'built from corners' and 'carved by walls.'
A tetrahedron as a convex hull of 4 points equals an intersection of 4 half-spaces.
Converting between the V- and H-descriptions is computationally expensive: a polytope given by m facets can have exponentially many vertices in high dimension (and vice versa). The two presentations are equivalent in principle but not cheaply interchangeable in practice.