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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 (geometry)|hedgehog]], a type of curve determined as the [[Envelope (mathematics)|envelope]] 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.{{r|martinez-maure}}
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 (geometry)|hedgehog]], a type of curve determined as the [[Envelope (mathematics)|envelope]] 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.{{r|martinez-maure}}


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.{{r|gray}}{{efn|There exist smooth open curves that do not have three parallel tangents but are not convex; the [[witch of Agnesi]] is an example.}}
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.{{r|gray}}{{efn|There exist smooth open curves that do not have three parallel tangents but are not convex; the graph of any cubic polynomial is an example. For the graph of a function, the slope of any tangent line is the derivative of the function at that point,{{r|abramson}} and since the derivative of a cubic is a quadratic polynomial, it produces any given slope at most twice.{{r|higgins}}}}


===Curvature===
===Curvature===
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==References==
==References==
{{reflist|refs=
{{reflist|refs=

<ref name=abramson>{{citation
| last = Abramson | first = Jay
| contribution = 3.1 Defining the derivative
| contribution-url = https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative
| publisher = [[OpenStax]]
| title = Precalculus
| url = https://openstax.org/details/books/precalculus
| year = 2014}}</reF>


<ref name=akopyan-avvakumov>{{citation
<ref name=akopyan-avvakumov>{{citation
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| volume = 122
| volume = 122
| year = 2010}}</ref>
| year = 2010}}</ref>

<ref name=higgins>{{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}}</ref>


<ref name=hug-weil>{{citation
<ref name=hug-weil>{{citation

Revision as of 07:08, 6 November 2022

Not to be confused with Convex function.
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. Examples of convex curves include the boundaries of convex sets and the graphs of convex functions.

Important subclasses of convex curves include the closed convex curves (the boundaries of 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. Combinations of these properties have also been considered.

Definitions

Among the plane curves, convex curves can be defined in many equivalent ways. Werner Fenchel credits Archimedes, in his On the Sphere and Cylinder, with the definition that they are the plane curves all of whose chords touch the same side of the curve.[1] 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 necessarily closed have also been called convex arcs.[2]

Background concepts

A plane curve is the image of any continuous function from an interval to the Euclidean plane. 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.[3]

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.[3] 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.[4] However, this terminology is ambiguous as other sources refer to a curve with two distinct endpoints as an open curve.[5] Here, we use the topological-line meaning of an open curve.

There are multiple definitions of smooth curves, involving the derivatives 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. If, in addition, the second derivative exists everywhere, then each of these points has a well-defined curvature.[3]

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-planes bounded by the line. A plane curve is called convex if it has a supporting line through each of its points.[6][7] For example, the graph of a convex function 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 tangent line.[8]

Supporting lines and tangent lines are not the same thing,[9] but for convex curves, every tangent line is a supporting line.[6] At a point of a curve where a tangent line exists, there can only be one supporting line, the tangent line.[10] 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 smooth curves.[11][a]

Boundaries of convex sets

A convex curve may be alternatively defined as a connected subset of the boundary of a convex set in the Euclidean plane.[6][7] Not every convex set has a connected boundary,[b] 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.[14] By the Jordan curve theorem, 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.[7][15] 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.[16]

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. Every connected subset of the boundary of a convex set has a support line through each of its points.[6][7][17]

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 curves (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 Euclidean plane to certain other linear spaces, with the same property that every point belongs to a supporting line.[18]

Strict convexity

The strictly convex curves again have many equivalent definitions. They are the convex curves that do not contain any line segments.[19] They are the curves for which every intersection of the curve with a line consists of at most two points.[18] They are the curves that can be formed as a connected subset of the boundary of a strictly convex set.[20] Here, a set is strictly convex if every point of its boundary is an extreme point of the set, the unique maximizer of some linear function.[21] 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 of any other subset of its points.[22]

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.[23][c]

Properties

Length and area

Every bounded convex curve is a rectifiable curve, meaning that has a well-defined finite length, and can be approximated in length by a sequence of inscribed polygonal chains. For closed convex curves, the length may be given by a form of the Crofton formula as times the expected value of the length of a projection of the curve onto a randomly-oriented line.[6] It is also possible to approximate the area of the convex hull of a convex curve by a sequence of inscribed convex polygons. For any integer , the most accurate approximating -gon has the property that each vertex has a supporting line parallel to the line through its two neighboring vertices.[25]

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.[26]

A smooth convex curve through 13 integer lattice points

It is not possible for a short strictly convex curve to pass through many points of the integer lattice. If the curve has length , then according to a theorem of Vojtěch Jarník, the number of lattice points that it can pass through is at most

as expressed in big O notation. The bound cannot be improved as there exist smooth strictly convex curves through this many points.[27]

Supporting lines and support function

A convex curve can have at most a countable set 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 the curve.[8][28] It is also possible to construct convex curves for which the singular points are dense.[17]

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 type of curve determined as the envelope 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.[29]

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.[11][d]

Curvature

An ellipse (red) and its evolute (blue), the locus of its centers of curvature. The four marked vertices of the ellipse correspond to the four cusps of the evolute.

According to the four-vertex theorem, every smooth closed curve has at least four vertices, points that are local minima or local maxima of curvature.[32] The original proof of the theorem, by Syamadas Mukhopadhyaya in 1909, considered only convex curves;[33] it was later extended to all smooth closed curves.[32]

Curvature can be used to characterize the smooth closed curves that are convex.[11] 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.[3] 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.[11][e] For strictly convex curves, although the curvature does not change sign, it may reach zero.[35]

The total absolute curvature of a smooth convex curve,

is at most . It is exactly 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 than , 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 is at least , with equality only for convex plane curves.[36][37]

Related shapes

An oval with a horizontal axis of symmetry

Smooth closed convex curves with an axis of symmetry, such as an ellipse or Moss's egg, may sometimes be called ovals.[38] However, in finite projective geometry, 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.[18]

The boundary of any convex polygon forms a convex curve (one that is a piecewise linear curve and not strictly convex). A polygon that is inscribed in any strictly convex curve, with its vertices in order along the curve, must be a convex polygon.[39] A scaled and rotated copy of any rectangle or trapezoid can be inscribed in any given closed convex curve. When the curve is smooth, a scaled and rotated copy of any cyclic quadrilateral can be inscribed in it.[40][41]

See also

Notes

  1. ^ The assumption of smoothness is necessary when defining convex curves using tangent lines. There exist fractal curves, and even the graphs of continuous functions, that do not have any tangent lines, not even vertical or one-sided tangents.[12] For these curves it is vacuously true that they lie on one side of each tangent line, but they are not convex.
  2. ^ For a slab, the region between two parallel lines, the boundary is its two defining lines.[13]
  3. ^ Many spirals are also locally convex but do not form closed curves.[7][24] Non-convex polygons are closed curves that are locally equivalent to the graphs of piecewise linear convex functions, but these functions are not strictly convex.
  4. ^ There exist smooth open curves that do not have three parallel tangents but are not convex; the graph of any cubic polynomial is an example. For the graph of a function, the slope of any tangent line is the derivative of the function at that point,[30] and since the derivative of a cubic is a quadratic polynomial, it produces any given slope at most twice.[31]
  5. ^ Some non-simple closed curves such as the rose curves also have consistently-signed curvatures.[34]

References

  1. ^ Fenchel, W. (1983), "Convexity through the ages", in Gruber, Peter M.; Wills, Jörg M. (eds.), Convexity and its Applications, Basel: Birkhäuser, pp. 120–130, doi:10.1007/978-3-0348-5858-8_6, MR 0731109
  2. ^ Strantzen, John; Brooks, Jeff (1992), "A chord-stretching map of a convex loop is an isometry", Geometriae Dedicata, 41 (1): 51–62, doi:10.1007/BF00181542, MR 1147501
  3. ^ a b c d Banchoff, Thomas F.; Lovett, Stephen T. (2016), "Chapter 1: Plane curves: local properties", Differential Geometry of Curves and Surfaces (2nd ed.), CRC Press, pp. 1–46, ISBN 978-1-4822-4737-4
  4. ^ Moore, Robert L. (1920), "Concerning simple continuous curves", Transactions of the American Mathematical Society, 21 (3): 333–347, doi:10.2307/1988935, JSTOR 1988935, MR 1501148
  5. ^ Veblen, Oswald (1931), The Cambrige Colloquium, 1916, Part. II: Analysis Situs, Colloquium Lectures, vol. 5, Providence, Rhode Island: American Mathematical Society, p. 3
  6. ^ a b c d e Toponogov, Victor A. (2006), "1.5 Problems: convex plane curves", Differential Geometry of Curves and Surfaces: A Concise Guide, Birkhäuser, pp. 15–19, doi:10.1007/b137116, ISBN 978-0-8176-4402-4
  7. ^ a b c d e Latecki, Longin Jan; Rosenfeld, Azriel (March 1998), "Supportedness and tameness differentialless geometry of plane curves" (PDF), Pattern Recognition, 31 (5): 607–622, doi:10.1016/s0031-3203(97)00071-x
  8. ^ a b Bourbaki, Nicolas (2004), Functions of a Real Variable: Elementary Theory, Elements of Mathematics, translated by Spain, Philip, Berlin: Springer-Verlag, p. 29, doi:10.1007/978-3-642-59315-4, ISBN 3-540-65340-6, MR 2013000
  9. ^ Rademacher, Hans; Toeplitz, Otto (1994), The Enjoyment of Math, Princeton Science Library, Princeton University Press, Princeton, NJ, p. 164, ISBN 0-691-02351-4, MR 1300411
  10. ^ Epstein, Charles L. (2008), Introduction to the Mathematics of Medical Imaging (2nd ed.), Society for Industrial and Applied Mathematics, p. 17, ISBN 978-0-89871-779-2
  11. ^ a b c d Gray, Alfred; Abbena, Elsa; Salamon, Simon (2006), "6.4 Convex plane curves", Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd ed.), Boca Raton: CRC Press, pp. 164–166, ISBN 978-1-58488-448-4; note that (per Definition 1.5, p. 5) this source assumes that the curves it describes are all piecewise smooth.
  12. ^ Ciesielski, Krzysztof Chris (2022), "Continuous maps admitting no tangent lines: a centennial of Besicovitch functions", The American Mathematical Monthly, 129 (7): 647–659, doi:10.1080/00029890.2022.2071562, MR 4457737
  13. ^ Preparata, Franco P.; Shamos, Michael Ian (1985), "2.2.2.1 The slab method", Computational Geometry: An Introduction, New York: Springer, pp. 45–48, doi:10.1007/978-1-4612-1098-6
  14. ^ Latecki, Longin Jan (1998), "Basic Definitions and Propositions", Discrete Representation of Spatial Objects in Computer Vision, Computational Imaging and Vision, vol. 11, Springer Netherlands, pp. 33–43, doi:10.1007/978-94-015-9002-0_2
  15. ^ Banchoff & Lovett (2016), p. 65.
  16. ^ Brinkhuis, Jan (2020), "Convex functions: basic properties", Convex Analysis for Optimization, Graduate Texts in Operations Research, Springer International Publishing, pp. 123–149, doi:10.1007/978-3-030-41804-5_5, ISBN 978-3-030-41804-5
  17. ^ a b Hug, Daniel; Weil, Wolfgang (2020), Lectures on Convex Geometry, Graduate Texts in Mathematics, vol. 286, Cham: Springer, doi:10.1007/978-3-030-50180-8, ISBN 978-3-030-50180-8, MR 4180684; see in particular Theorem 1.16 (support theorem), p. 27, and exercise 16, p. 60
  18. ^ a b c Polster, Burkard; Steinke, Günter (2001), "2.2.1 Convex curves, arcs, and ovals", Geometries on surfaces, Encyclopedia of Mathematics and its Applications, vol. 84, Cambridge University Press, pp. 31–34, doi:10.1017/CBO9780511549656, ISBN 0-521-66058-0, MR 1889925
  19. ^ Maehara, Hiroshi (2015), "Circle lattice point problem, revisited", Discrete Mathematics, 338 (3): 164–167, doi:10.1016/j.disc.2014.11.004, MR 3291879
  20. ^ Hartmann, Erich; Feng, Yu Yu (1993), "On the convexity of functional splines", Computer Aided Geometric Design, 10 (2): 127–142, doi:10.1016/0167-8396(93)90016-V, MR 1213308
  21. ^ Ha, Truong Xuan Duc; Jahn, Johannes (2019), "Characterizations of strictly convex sets by the uniqueness of support points", Optimization, 68 (7): 1321–1335, doi:10.1080/02331934.2018.1476513, MR 3985200
  22. ^ Garibaldi, Julia; Iosevich, Alex; Senger, Steven (2011), The Erdős Distance Problem, Student Mathematical Library, vol. 56, Providence, Rhode Island: American Mathematical Society, p. 51, doi:10.1090/stml/056, ISBN 978-0-8218-5281-1, MR 2721878
  23. ^ Ricci, Fulvio; Travaglini, Giancarlo (2001), "Convex curves, Radon transforms and convolution operators defined by singular measures", Proceedings of the American Mathematical Society, 129 (6): 1739–1744, doi:10.1090/S0002-9939-00-05751-8, MR 1814105
  24. ^ Umehara, Masaaki; Yamada, Kotaro (2017), "Chapter 4: Geometry of spirals", Differential Geometry of Curves and Surfaces, Hackensack, New Jersey: World Scientific Publishing, pp. 40–49, doi:10.1142/9901, ISBN 978-981-4740-23-4, MR 3676571
  25. ^ Johnson, Harold H.; Vogt, Andrew (1980), "A geometric method for approximating convex arcs", SIAM Journal on Applied Mathematics, 38 (2): 317–325, doi:10.1137/0138027, MR 0564017
  26. ^ Arnold, V. I. (1989), "Topological proof of the transcendence of the abelian integrals in Newton's Principia", Istoriko-Matematicheskie Issledovaniya (31): 7–17, ISSN 0136-0949, MR 0993175
  27. ^ Bordellès, Olivier (2012), "5.4.7 Counting integer points on smooth curves", Arithmetic Tales, Springer, p. 290, ISBN 9781447140962
  28. ^ Kakeya, Sōichi (1915), "On some properties of convex curves and surfaces.", Tohoku Mathematical Journal, 8: 218–221, JFM 45.1348.02
  29. ^ Martinez-Maure, Yves (2001), "A fractal projective hedgehog", Demonstratio Mathematica, 34 (1): 59–63, doi:10.1515/dema-2001-0108, MR 1823083
  30. ^ Abramson, Jay (2014), "3.1 Defining the derivative", Precalculus, OpenStax
  31. ^ Higgins, Peter M. (2008), Number Story: From Counting to Cryptography, London: Springer, p. 179, doi:10.1007/978-1-84800-001-8
  32. ^ a b DeTruck, Dennis; Gluck, Herman; Pomerleano, Daniel; Vick, David Shea (2007), "The four vertex theorem and its converse" (PDF), Notices of the American Mathematical Society, 54 (2): 9268, arXiv:math/0609268
  33. ^ Mukhopadhyaya, S. (1909), "New methods in the geometry of a plane arc", Bulletin of the Calcutta Mathematical Society, 1: 21–27
  34. ^ Cieślak, Waldemar; Zając, Józef (1986), "The rosettes", Mathematica Scandinavica, 58 (1): 114–118, doi:10.7146/math.scand.a-12133, JSTOR 24491607, MR 0845490
  35. ^ Helton, J. William; Nie, Jiawang (2010), "Semidefinite representation of convex sets", Mathematical Programming, 122 (1, Ser. A): 21–64, doi:10.1007/s10107-008-0240-y, MR 2533752
  36. ^ Chen, Bang-Yen (2000), "Riemannian submanifolds", Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, pp. 187–418, doi:10.1016/S1874-5741(00)80006-0, MR 1736854; see in particular p. 360
  37. ^ Banchoff & Lovett (2016), p. 108.
  38. ^ Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, MAA Spectrum, Mathematical Association of America, p. 156, ISBN 9780883855119
  39. ^ Milnor, J. W. (1950), "On the total curvature of knots", Annals of Mathematics, Second Series, 52: 248–257, doi:10.2307/1969467, JSTOR 1969467, MR 0037509; see discussion following Theorem 3.4 (Fenchel's theorem), p. 254
  40. ^ Akopyan, Arseniy; Avvakumov, Sergey (2018), "Any cyclic quadrilateral can be inscribed in any closed convex smooth curve", Forum of Mathematics, 6: Paper No. e7, 9, doi:10.1017/fms.2018.7, MR 3810027
  41. ^ Matschke, Benjamin (2021), "Quadrilaterals inscribed in convex curves", Transactions of the American Mathematical Society, 374 (8): 5719–5738, doi:10.1090/tran/8359, MR 4293786
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