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An example of a convex polygon: a regular pentagon. In 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). [1]
Convex curve. 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. A parabola, a convex curve that is the graph of the convex function. In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). [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.
Polygon. Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting. In geometry, a polygon (/ ˈpɒlɪɡɒn /) is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal ...
Convex cone. A convex cone (light blue). Inside of it, the light red convex cone consists of all points αx + βy with α, β > 0, for the depicted x and y. The curves on the upper right symbolize that the regions are infinite in extent. In linear algebra, a cone —sometimes called a linear cone for distinguishing it from other sorts of cones ...
The problem of counting integer points in convex polyhedra arises in several areas of mathematics and computer science. [25] In application areas, the dot planimeter is a transparency-based device for estimating the area of a shape by counting the grid points that it contains. [26]
The convex hull of the red set is the blue and red convex set. In geometry, the convex hull, convex envelope or convex closure[1] 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 ...
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
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