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Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve.
9-cube. In geometry, a 9-cube is a nine- dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces . It can be named by its Schläfli symbol {4,3 7 }, being composed of three 8-cubes around each 7-face.
Convex geometry. In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.
Mixed volume. In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in . This number depends on the size and shape of the bodies, and their relative orientation to each other.
In geometry, a cube [a] is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross. [1] The cube is the only regular hexahedron and is one of the five Platonic solids.
Catalan solid. A toroidal polyhedron. In geometry, a polyhedron ( pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many', and ἕδρον (-hedron) 'base, seat') is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices . A convex polyhedron is a polyhedron that bounds a convex set.
Rectified 9-cubes. In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube . There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex.
The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula. where V, E, and F are respectively the numbers of v ertices (corners), e dges and f aces in the given polyhedron. Any convex polyhedron 's surface has Euler characteristic. This equation, stated by Euler in 1758, [2] is known as Euler's ...