geometry Geometry (; ) 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 w ...

, a convex curve is a

plane curve In mathematics, a plane curve is a curve in a plane that may be 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 plane c ...

that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

. 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 ...

s, the boundaries of

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

s, and the graphs of

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

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

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

within the curve, and the distance of these lines from the origin defines a continuous support function. A smooth simple closed curve is convex if and only if its

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

has a consistent sign, which happens if and only if its total curvature equals its total absolute curvature.

Definitions

, in his ''

On the Sphere and Cylinder ''On the Sphere and Cylinder'' () is a treatise that was published by Archimedes in two volumes . It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and w ...

'', defines convex arcs as the plane curves that lie on one side of the line through their two endpoints, and for which all chords 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. For the next two millennia, there was little study of convexity: its in-depth investigation began again only in the 19th century, when

Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

and others began using

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

instead of algebraic methods to put

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

on a more rigorous footing. 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 small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

from an interval to the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

. Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be

continuously differentiable In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...

, and in some contexts are defined to require higher derivatives. The function parameterizing a smooth curve is often assumed to be regular, meaning that its derivative stays away from zero; intuitively, the moving point never slows to a halt or reverses direction. Each interior point of a smooth curve has a

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

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

. 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.

Supporting lines

A supporting line is a line containing at least one point of the curve, for which the curve is contained in one of the two

half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

s 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 discret ...

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 function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be ...

smooth curves.

Boundaries of convex sets

A convex curve may be alternatively defined as a connected subset of the boundary 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 (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be ...

, 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, 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 ...

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 such as the

real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

. In these spaces, like in the Euclidean plane, any curve with only these restricted line intersections has a supporting line for each point.

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 line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

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. 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 number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...

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, meaning that none of their points can be a

convex combination In convex geometry and Vector space, vector algebra, a convex combination is a linear combination of point (geometry), points (which can be vector (geometric), vectors, scalar (mathematics), scalars, or more generally points in an affine sp ...

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. This means that, at each point of the curve, there is a

neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...

of the points and a system of

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

within that neighborhood such that, within that neighborhood, the curve coincides with the graph of a strictly convex function.

Symmetry

Smooth closed convex curves with an

axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...

, 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 ...

or Moss's egg, 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 of mathematics (projective geometry, technical drawing, etc.), it is given a more precise definition, which may inc ...

s. However, the same word has also been used to describe the sets for which each point has a unique line disjoint from the rest of the set, especially in the context of ovals 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 (''p ...

. In Euclidean geometry these are the smooth strictly convex closed curves, without any requirement of symmetry.

Properties

Length and area

Every bounded convex curve is a rectifiable curve, meaning that it has a well-defined finite

arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

, 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 as

\pi

times the average length of its projections onto lines. 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, the area cut off from an infinitely differentiable 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 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 (group), lattice in the Euclidean space whose lattice points are tuple, -tuples of integers. The two-dimensional integer lattice is also called the s ...

. If the curve has length

L

, then according to a theorem of

Vojtěch Jarník Vojtěch Jarník (; 22 December 1897 – 22 September 1970) was a Czech mathematician. He 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 ...

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

\fracL^+O(L^).

Because this estimate uses

big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...

, it is accurate only in the limiting case of large lengths. Neither the leading constant nor the exponent in the error term can be improved.

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 numbe ...

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 ...

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 support function, 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 17 species of hedgehog in five genera found throughout parts of Europe, Asia, and Africa, and in New Zealand by introduction. The ...

, 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 (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...

of a system of lines with a continuous support function. The hedgehogs also include non-convex curves, such as the astroid, and even self-crossing curves, but the smooth strictly convex curves are the only hedgehogs that have no singular points. It is impossible for a convex curve to have three parallel tangent lines. More strongly, a smooth closed curve is convex if and only if it does not have three parallel tangent lines. In one direction, the middle of any three parallel tangent lines would separate the points of tangency of the other two lines, so it could not be a line of support. There could be no other line of support through its point of tangency, so a curve tangent to these three lines could not be convex. In the other direction, a non-convex smooth closed curve has at least one point with no support line. The tangent line through that point, and the two tangent supporting lines parallel to it, form a set of three parallel tangent lines.

Curvature

According to the

four-vertex theorem In geometry, the four-vertex theorem 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 ...

, 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 in 1909, considered only convex it was later extended to all smooth closed Curvature can be used to characterize the smooth closed curves that are 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 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 Every smooth simple closed curve with strictly positive (or strictly negative) curvature is strictly convex, but some strictly convex curves can have points with curvature The total absolute curvature of a smooth convex curve,

\int, \kappa(s), ds,

is at It is exactly

2\pi

for closed convex curves, equalling the total curvature 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 and its excess can be used as a measure of how far from convex the curve is. More generally, by Fenchel's theorem, 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 ...

is at with equality only for convex plane By the Alexandrov theorem, a non-smooth convex curve has a second derivative, and therefore a well-defined curvature,

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

. This means that the subset of points without a second derivative has

measure zero In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...

in the curve. However, in other senses, the set of points with a second derivative can be small. In particular, for the graphs of generic non-smooth convex functions, it is a

meager set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...

, that is, a countable union of

nowhere dense set In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...

Inscribed polygons

The boundary of any

forms a convex curve (one that is a piecewise linear curve and not strictly convex). A polygon that is

inscribed 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 figure G" means precisely the same th ...

in any strictly convex curve, with its vertices in order along the curve, must be a convex polygon. The

inscribed square problem The inscribed square problem, also known as the square peg problem or the Toeplitz conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is true if the curve is ...

is the problem of proving that every simple closed curve in the plane contains the four corners of a square. Although still unsolved in general, its solved cases include the convex curves. In connection with this problem, related problems of finding inscribed quadrilaterals have been studied for convex curves. A scaled and rotated copy of any

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

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

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

can be inscribed in it. However, the assumption of smoothness is necessary for this result, because some

right kite In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eucl ...

s cannot be inscribed in some obtuse

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

Regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s with more than four sides cannot be inscribed in all closed convex curves, because the curve formed by a

semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetr ...

and its diameter does not contain any of these polygons.

Notes

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American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, location = Providence, Rhode Island , series = Student Mathematical Library , title = The Erdős Distance Problem , title-link = The Erdős Distance Problem , volume = 56 , year = 2011 {{citation , last1 = Gray , first1 = Alfred , author1-link = Alfred Gray (mathematician) , last2 = Abbena , first2 = Elsa , last3 = Salamon , first3 = Simon , contribution = 6.4 Convex plane curves , edition = 3rd , isbn = 978-1-58488-448-4 , location = Boca Raton, Florida , pages = 164–166 , publisher =

CRC Press The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technol ...

, title = Modern Differential Geometry of Curves and Surfaces with Mathematica , year = 2006; note that (per Definition 1.5, p. 5) this source assumes that the curves it describes are all piecewise smooth. {{citation , last = Gruber , first = Peter M. , author-link = Peter M. Gruber , contribution = 2.2: Alexandrov's theorem in second-order differentiability , isbn = 978-3-540-71132-2 , mr = 2335496 , pages = 27–32 , publisher = Springer , location = Berlin , series = Grundlehren der Mathematischen Wissenschaften undamental Principles of Mathematical Sciences , title = Convex and Discrete Geometry , volume = 336 , year = 2007 {{citation , last = Gruber , first = Peter M. , author-link = Peter M. Gruber , editor-last = Gruber , editor-first = Peter M. , contribution = History of convexity , contribution-url = https://books.google.com/books?id=M2viBQAAQBAJ&pg=PA1 , isbn = 0-444-89598-1 , pages = 1–15 , publisher = North-Holland , location = Amsterdam , title = Handbook of Convex Geometry, Volume A , year = 1993 {{citation , last1 = Ha , first1 = Truong Xuan Duc , last2 = Jahn , first2 = Johannes , doi = 10.1080/02331934.2018.1476513 , issue = 7 , journal = Optimization , mr = 3985200 , pages = 1321–1335 , title = Characterizations of strictly convex sets by the uniqueness of support points , volume = 68 , year = 2019, s2cid = 126177709 {{citation , last1 = Hartmann , first1 = Erich , last2 = Feng , first2 = Yu Yu , doi = 10.1016/0167-8396(93)90016-V , issue = 2 , journal = Computer Aided Geometric Design , mr = 1213308 , pages = 127–142 , title = On the convexity of functional splines , volume = 10 , year = 1993 {{citation , last1 = Helton , first1 = J. William , last2 = Nie , first2 = Jiawang , doi = 10.1007/s10107-008-0240-y , issue = 1, Ser. A , journal =

Mathematical Programming Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

, mr = 2533752 , pages = 21–64 , title = Semidefinite representation of convex sets , volume = 122 , year = 2010, arxiv = 0705.4068 , s2cid = 1352703 {{citation , last = Higgins , first = Peter M. , doi = 10.1007/978-1-84800-001-8 , location = London , page = 179 , publisher = Springer , title = Number Story: From Counting to Cryptography , url = https://books.google.com/books?id=HcIwkWXy3CwC&pg=PA179 , year = 2008, isbn = 978-1-84800-000-1 {{citation , last1 = Hug , first1 = Daniel , last2 = Weil , first2 = Wolfgang , doi = 10.1007/978-3-030-50180-8 , isbn = 978-3-030-50180-8 , mr = 4180684 , publisher = Springer , location = Cham , series =

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...

, title = Lectures on Convex Geometry , volume = 286 , year = 2020, s2cid = 226548863 ; see in particular Theorem 1.16 (support theorem), p. 27, and exercise 16, p. 60 {{citation , last = Jerrard , first = R. P. , doi = 10.1090/s0002-9947-1961-0120604-3 , issue = 2 , journal =

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

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Pattern Recognition Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their p ...

, pages = 607–622 , title = Supportedness and tameness differentialless geometry of plane curves , url = https://cis.temple.edu/~latecki/Papers/pr98.pdf , volume = 31, bibcode = 1998PatRe..31..607L {{citation , last = Maehara , first = Hiroshi , doi = 10.1016/j.disc.2014.11.004 , issue = 3 , journal =

Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

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Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

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Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

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Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

, location=Princeton, New Jersey , series = Princeton Science Library , title = The Enjoyment of Math , url = https://books.google.com/books?id=Y_9ZDwAAQBAJ&pg=PA164 , year = 1994 {{citation , last1 = Ricci , first1 = Fulvio , last2 = Travaglini , first2 = Giancarlo , doi = 10.1090/S0002-9939-00-05751-8 , issue = 6 , journal = Proceedings of the American Mathematical Society , mr = 1814105 , pages = 1739–1744 , title = Convex curves, Radon transforms and convolution operators defined by singular measures , volume = 129 , year = 2001, doi-access = free {{citation, title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, series=MAA Spectrum, first=Steven, last=Schwartzman, publisher=Mathematical Association of America, year=1994, isbn=9780883855119, pag
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Journal of Number Theory The ''Journal of Number Theory'' (''JNT'') is a monthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State Univ ...

, mr = 337857 , pages = 128–135 , title = The number of lattice points on a convex curve , volume = 6 , year = 1974, issue = 2 , bibcode = 1974JNT.....6..128S , doi-access = free {{citation , last = Stromquist , first = Walter , doi = 10.1112/S0025579300013061 , issue = 2 , journal =

Mathematika ''Mathematika'' is a peer-reviewed mathematics journal that publishes both pure and applied mathematical articles. The journal was founded by Harold Davenport in the 1950s. The journal is published by the London Mathematical Society, on behalf of ...

, mr = 1045781 , pages = 187–197 , title = Inscribed squares and square-like quadrilaterals in closed curves , volume = 36 , year = 1989 {{citation , last = Toponogov , first = Victor A. , editor-first1 = Vladimir Y , editor-last1 = Rovenski , author-link = Victor Andreevich Toponogov , contribution = 1.5 Problems: convex plane curves , doi = 10.1007/b137116 , isbn = 978-0-8176-4402-4 , pages = 15–19 , publisher = Birkhäuser , title = Differential Geometry of Curves and Surfaces: A Concise Guide , year = 2006 {{citation , last1 = Umehara , first1 = Masaaki , last2 = Yamada , first2 = Kotaro , contribution = Chapter 4: Geometry of spirals , contribution-url = https://books.google.com/books?id=tlEyDwAAQBAJ&pg=PA40 , doi = 10.1142/9901 , isbn = 978-981-4740-23-4 , mr = 3676571 , pages = 40–49 , publisher =

World Scientific Publishing World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, with more than 170 journals in var ...

, location = Hackensack, New Jersey , title = Differential Geometry of Curves and Surfaces , year = 2017 {{citation , last = Veblen , first = Oswald , author-link = Oswald Veblen , location = Providence, Rhode Island , page = 3 , publisher = American Mathematical Society , series = Colloquium Lectures , title = The Cambridge Colloquium, 1916, Part. II: Analysis Situs , url = https://books.google.com/books?id=jq2XYR6of4oC&pg=PA3 , volume = 5 , year = 1931 {{citation , last = Yurinsky , first = Vadim Vladimirovich , contribution = 1.4.4 Piecewise-linear functions and polytopes , contribution-url = https://books.google.com/books?id=o_d6CwAAQBAJ&pg=PA24 , doi = 10.1007/bfb0092599 , location = Berlin & Heidelberg , pages = 24–27 , publisher = Springer , title = Sums and Gaussian Vectors , series = Lecture Notes in Mathematics , year = 1995, volume = 1617 , isbn = 978-3-540-60311-5 Convex geometry Plane curves