A convex polytope is a geometric shape in multi-dimensional space defined by a finite number of flat surfaces, known as faces. Each face is a polygon, and the shape is characterized by the property that any line segment connecting two points within the polytope lies entirely inside it. This makes convex polytopes important in various fields, including mathematics and computer science. Convex polytopes can exist in any number of dimensions, with the simplest example being a three-dimensional shape like a cube or a tetrahedron. They are studied in the field of convex geometry and have applications in optimization problems, such as those found in linear programming.