geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a convex curve is a

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...

that has a

supporting line In geometry, a supporting line ''L'' of a curve ''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemannp. 158/ref> In other words, ''C'' lies completely ...

through each of its points. There are many other equivalent definitions of these curves, going back to

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

. Examples of convex curves include the

convex polygon 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 ...

s, the boundaries of

convex set 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 r ...

s, and the

graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...

s. Important subclasses of convex curves include the closed convex curves (the boundaries of bounded convex sets), the

smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s that are convex, and the strictly convex curves, which have the additional property that each supporting line passes through a unique point of the curve. Combinations of these properties have also been considered. Bounded convex curves have a well-defined length, which can be obtained by approximating them with polygons, or from the average length of their projections onto a line. The maximum number of grid points that can belong to a single curve is controlled by its length. The points at which a convex curve has a unique

are

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

within the curve, and the distance of these lines from the origin defines a continuous

support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any n ...

. A smooth closed curve is convex if and only if its

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

has a consistent sign, which happens if and only if its

total curvature In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: :\int_a^b k(s)\,ds. The total curvature of a closed curve i ...

equals its

total absolute curvature In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of ...

Definitions

Werner Fenchel Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theor ...

credits

, in his ''

On the Sphere and Cylinder ''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of t ...

'', with the definition of convex arcs as the plane curves that lie on one side of the line through their two endpoints, and for which all

chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...

touch the same side of the curve. This may have been the first formal definition of any notion of convexity, although convex polygons and convex polyhedra were already long known before Archimedes. There was little study of convexity after Archimedes for many centuries, until the development of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

led to both renewed interest in the subject and tools for approaching it. Many other equivalent definitions for the convex curves are possible, as detailed below. Convex curves have also been defined by their supporting lines, by the sets they form boundaries of, and by their intersections with lines. In order to distinguish closed convex curves from curves that are not closed, the closed convex curves have sometimes also been called ''convex loops'', and convex curves that are not closed have also been called ''convex arcs''.

Background concepts

is the image of any

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

from an interval to the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

. Intuitively, it is a set of points that could be traced out by a moving point. Often, the function used to describe this motion is required not only to be continuous, but also ''regular'', meaning that the moving point never slows to a halt or reverses direction. A plane curve is ''closed'' if the two endpoints of the interval are mapped to the same point in the plane, and it is ''simple'' if no other two points coincide. Less commonly, a simple plane curve may be said to be ''open'' if it is topologically equivalent to a line, neither having an endpoint nor forming any limiting point that does not belong to it, and dividing the plane into two unbounded regions. However, this terminology is ambiguous as other sources refer to a curve with two distinct endpoints as an open curve. Here, we use the topological-line meaning of an open curve. There are multiple definitions of

s, involving the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

s of the function defining the curve. If it is regular and has a derivative everywhere, then each interior point of the curve has a

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

. If, in addition, the second derivative exists everywhere, then each of these points has a well-defined

Supporting lines

is a line containing at least one point of the curve, for which the curve is contained in one of the two half-planes bounded by the line. A plane curve is called ''convex'' if it has a supporting line through each of its points. For example, the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

of a

has a supporting line below the graph through each of its points. More strongly, at the points where the function has a derivative, there is exactly one supporting line, the

. Supporting lines and tangent lines are not the same thing, but for convex curves, every tangent line is a supporting line. At a point of a curve where a tangent line exists, there can only be one supporting line, the tangent line. Therefore, a smooth curve is convex if it lies on one side of each of its tangent lines. This may be used as an equivalent definition of convexity for smooth curves, or more generally for

piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...

smooth curves.

Boundaries of convex sets

A convex curve may be alternatively defined as a connected subset of the

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...

of a

in the

. Not every convex set has a connected boundary, but when it does, the whole boundary is an example of a convex curve. When a bounded convex set in the plane is not a line segment, its boundary forms a simple closed convex curve. By the

Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...

, a simple closed curve divides the plane into interior and exterior regions, and another equivalent definition of a closed convex curve is that it is a simple closed curve whose union with its interior is a convex set. Examples of open and unbounded convex curves include the graphs of convex functions. Again, these are boundaries of convex sets, the epigraphs of the same functions. This definition is equivalent to the definition of convex curves from support lines. Every convex curve, defined as a curve with a support line through each point, is a subset of the boundary of its own

convex hull In 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 ...

. Every connected subset of the boundary of a convex set has a support line through each of its points.

Intersection with lines

For a convex curve, every line in the plane intersects the curve in one of four ways: its intersection can be the empty set, a single point, a pair of points, or an interval. In the cases where a closed curve intersects in a single point or an interval, the line is a supporting line. This can be used as an alternative definition of the convex curves: they are the

Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...

s (connected simple curves) for which every intersection with a line has one of these four types. This definition can be used to generalize convex curves from the

to certain other linear spaces, with the same property that every point belongs to a supporting line.

Strict convexity

The strictly convex curves again have many equivalent definitions. They are the convex curves that do not contain any

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

s. They are the curves for which every intersection of the curve with a line consists of at most two points. They are the curves that can be formed as a connected subset of the boundary of a

strictly convex set 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 r ...

. Here, a set is strictly convex if every point of its boundary is an

extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ...

of the set, the unique maximizer of some linear function. As the boundaries of strictly convex sets, these are the curves that lie in

convex position In discrete and computational geometry, a set of points in the Euclidean plane or a higher-dimensional Euclidean space is said to be in convex position or convex independent if none of the points can be represented as a convex combination of the o ...

, meaning that none of their points can be a

convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...

of any other subset of its points. Closed strictly convex curves can be defined as the simple closed curves that are locally equivalent (under an appropriate coordinate transformation) to the graphs of strictly convex functions.

Properties

Length and area

Every bounded convex curve is a

rectifiable curve Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Rec ...

, meaning that has a well-defined finite

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

, and can be approximated in length by a sequence of inscribed

polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

s. For closed convex curves, the length may be given by a form of the

Crofton formula In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it. Statement Suppose \gamma is a ...

\pi

times the

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

of the length of a projection of the curve onto a randomly-oriented line. It is also possible to approximate the area of the convex hull of a convex curve by a sequence of inscribed

s. For any integer

n

, the most accurate approximating

n

-gon has the property that each vertex has a supporting line parallel to the line through its two neighboring vertices. As Archimedes already knew, if two convex curves have the same endpoint, and one of the two curves lies between the other and the line through their endpoints, then the inner curve is shorter than the outer one. According to

Newton's theorem about ovals In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI of book 1 of Newton's '' Pri ...

, the area cut off from an

infinitely differentiable In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...

convex curve by a line cannot be an algebraic function of the coefficients of the line. Convex curve through 13 grid points

It is not possible for a short strictly convex curve to pass through many points of the

integer lattice In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...

. If the curve has length

L

, then according to a theorem of

Vojtěch Jarník Vojtěch Jarník (; 1897–1970) was a Czech mathematician who worked for many years as a professor and administrator at Charles University, and helped found the Czechoslovak Academy of Sciences. He is the namesake of Jarník's algorithm for ...

, the number of lattice points that it can pass through is at most

\fracL^+O(L^),

as expressed in

big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...

. The bound cannot be improved as there exist smooth strictly convex curves through this many points.

Supporting lines and support function

A convex curve can have at most a

countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

of singular points, where it has more than one supporting line. All of the remaining points must be non-singular, and the unique supporting line at these points is necessarily a tangent line. This implies that the non-singular points form a

dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...

in the curve. It is also possible to construct convex curves for which the singular points are dense. A closed strictly convex closed curve has a continuous

, mapping each direction of supporting lines to their signed distance from the origin. It is an example of a

hedgehog A hedgehog is a spiny mammal of the subfamily Erinaceinae, in the eulipotyphlan family Erinaceidae. There are seventeen species of hedgehog in five genera found throughout parts of Europe, Asia, and Africa, and in New Zealand by introducti ...

, a type of curve determined as the

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...

of a system of lines with a continuous support function. The hedgehogs also include non-convex curves, such as the

astroid In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. By double generation, it ...

, and even self-crossing curves, but the smooth strictly convex curves are the only hedgehogs that have no singular points. Every curve has at most two supporting lines in each direction. For a bounded curve that does not lie on a single line of the same direction, there are exactly two. If a curve has three distinct parallel tangent lines, at least one of them cannot be a supporting line, there can be no other supporting line through the same point, and so the curve is not convex. If a smooth closed curve is non-convex, it has a point with no supporting line, and the tangent line at this point is parallel to two more tangent supporting lines. Therefore, a smooth closed curve is convex if and only if it does not have three parallel tangent lines.

Curvature

According to the

four-vertex theorem The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives fro ...

, every smooth closed curve has at least four vertices, points that are local minima or local maxima of

. The original proof of the theorem, by

Syamadas Mukhopadhyaya Syamadas Mukhopadhyaya (22 June 1866 – 8 May 1937) was an Indian mathematician who introduced the four-vertex theorem and Mukhopadhyaya's theorem in plane geometry. Biography Syamadas Mukhopadhyaya was born at Haripal, Hooghly district, ...

in 1909, considered only convex curves; it was later extended to all smooth closed curves. Curvature can be used to characterize the smooth closed curves that are convex. The curvature depends in a trivial way on the parameterization of the curve: if a regularly parameterization of a curve is reversed, the same set of points results, but its curvature is negated. A smooth simple closed curve, with a regular parameterization, is convex if and only if its curvature has a consistent sign: always non-negative, or always non-positive. For strictly convex curves, although the curvature does not change sign, it may reach zero. The

of a smooth convex curve,

\int, \kappa(s), ds,

is at most

2\pi

. It is exactly

2\pi

for closed convex curves, equalling the

of these curves, and of any simple closed curve. For convex curves, the equality of total absolute curvature and total curvature follows from the fact that the curvature has a consistent sign. For closed curves that are not convex, the total absolute curvature is always greater than

2\pi

, and its excess can be used as a measure of how far from convex the curve is. More generally, by

Fenchel's theorem In differential geometry, Fenchel's theorem is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least 2\pi. Equivalently, the average curvature is at least 2 \pi/L, where L is the length ...

, the total absolute curvature of a closed smooth

space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...

is at least

2\pi

, with equality only for convex plane curves.

Related shapes

Smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

closed convex curves with an

axis of symmetry Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.

, such as an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

Moss's egg In Euclidean geometry, Moss's egg is an oval made by smoothly connecting four circular arcs. It can be constructed from a right isosceles triangle ''ABC'' with apex ''C''. To construct Moss's egg: *Draw a semicircle In mathematics (and more sp ...

, may sometimes be called

oval An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...

s. However, in finite

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

, ovals are instead defined as sets for which each point has a unique line disjoint from the rest of the set, a property that in Euclidean geometry is true of the smooth strictly convex closed curves. The boundary of any

forms a convex curve (one that is a

piecewise linear curve In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

and not strictly convex). A polygon that is

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

in any strictly convex curve, with its vertices in order along the curve, must be a convex polygon. A scaled and rotated copy of any

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadri ...

can be inscribed in any given closed convex curve. When the curve is smooth, a scaled and rotated copy of any

cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

can be inscribed in it.

Notes

References

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