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Krein–Milman theorem: A convex polygon is the convex hull of its vertices. Thus it is fully defined by the set of its vertices, and one only needs the corners of the polygon to recover the entire polygon shape. Hyperplane separation theorem: Any two convex polygons with no points in common have a separator line. If the polygons are closed and ...
A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
Equivalently, a convex set or a convex region is a set that intersects every line in a line segment, single point, or the empty set. [1] [2] 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.
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. [131] It has close connections to convex analysis, optimization and functional analysis and important applications in number theory. Convex geometry dates back to antiquity. [131]
The weight of the large crowd caused the bridge to sag 7 feet, flattening its usual convex shape. Engineer Daniel E. Mohn reaffirmed the bridge was not overstressed as a result of Bridgewalk '87.
The regular tetrahedron is the simplest convex deltahedron, a polyhedron in which all of its faces are equilateral triangles; there are seven other convex deltahedra. [ 3 ] The regular tetrahedron is also one of the five regular Platonic solids , a set of polyhedrons in which all of their faces are regular polygons . [ 4 ]