Convex geometryIn 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. According to the Mathematics Subject Classification MSC2010, the mathematical discipline Convex and Discrete Geometry includes three major branches: general convexity polytopes and polyhedra discrete geometry (though only portions of the latter two are included in convex geometry).
Convex setIn 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). 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 is always a convex curve.
Convex hullIn geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact.
Convex polytopeA convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue.
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Convex polygonIn geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. A strictly convex polygon is a convex polygon such that no line contains two of its edges.
Isoperimetric inequalityIn mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume , where is a unit sphere. The equality holds only when is a sphere in . On a plane, i.e. when , the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means "having the same perimeter".
Minkowski additionIn geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B: The Minkowski difference (also Minkowski subtraction, Minkowski decomposition, or geometric difference) is the corresponding inverse, where produces a set that could be summed with B to recover A. This is defined as the complement of the Minkowski sum of the complement of A with the reflection of B about the origin. This definition allows a symmetrical relationship between the Minkowski sum and difference.
Mapie de Toulouse-LautrecMarie Pierre "Mapie" de Toulouse-Lautrec (1901–1972) was a French journalist and food writer, born Marie Pierre Adélaïde Lévêque de Vilmorin in Verrières-le-Buisson, scion of the Vilmorin seed company. Her horticulturalist father was Joseph Marie Philippe Lévêque de Vilmorin (1872-1917), and her mother was the former Bertha Marie Mélanie de Gaufridy de Dortan (1876-1937). The writer Louise de Vilmorin (1902–1969) was her younger sister, while one of her younger brothers, Roger, was the result of an affair between her mother and Alfonso XIII of Spain.
Philippe de VilmorinJoseph-Marie-Philippe Lévêque de Vilmorin (21 May 1872 – 29 June 1917), generally known as Philippe de Vilmorin, was a noted French botanist and plant collector, and a member of the celebrated Vilmorin family of horticulturists. In 1903 Vilmorin began the Arboretum de Pézanin, an arboretum located in Dompierre-les-Ormes, Saône-et-Loire, Bourgogne, France. He also collected plants in Egypt and Sudan that now form part of the herbarium of the National Botanic Garden of Belgium.
VilmorinVilmorin is a French seed producer. The company has a long history in France, where it was family-controlled for almost two centuries, and today exists as a publicly traded company owned principally by agro-industrial cooperative Groupe Limagrain, the largest plant breeding and seed company in the European Union. Vilmorin was founded as a plant and seed boutique in 1743 by seed expert Claude Geoffroy and her husband Pierre Andrieux, the chief seed supplier and botanist to King Louis XV.
11 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.
