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In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty).
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.
Regular polyhedron. A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular ...
Regular polygon. In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew.
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 ...
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes.
In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces in quantum cohomology. [1] : §1 [2] [3] These moduli spaces are smooth orbifolds whenever the target space is convex. A variety is called convex if the pullback of the tangent bundle to a ...
In elementary geometry, a face is a polygon [note 1] on the boundary of a polyhedron. [2] [3] Other names for a polygonal face include polyhedron side and Euclidean plane tile . For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope.
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