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Mirror with a curved reflecting surface
with a curved reflecting surface. The surface may be either convex (bulging outward) or concave (recessed inward). Most curved mirrors have surfaces that
Curved_mirror
Negative of a convex function
a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination
Concave_function
Real function with secant line between points above the graph itself
while a concave function's graph is shaped like a cap ∩ {\displaystyle \cap } . A twice-differentiable function of a single variable is convex if and only
Convex_function
Lithograph by Dutch artist M. C. Escher
Convex and Concave is a lithograph print by the Dutch artist M. C. Escher, first printed in March 1955. It depicts an ornate architectural structure with
Convex_and_Concave
Optical device which transmits and refracts light
of the other surface. A lens with one convex and one concave side is convex-concave or meniscus. Convex-concave lenses are most commonly used in corrective
Lens
Plane figure bounded by line segments
boundary of the polygon does not cross itself. All convex polygons are simple. Concave: Non-convex and simple. There is at least one interior angle greater
Polygon
In geometry, set whose intersection with every line is a single line segment
non-convex sets. A set that is not convex is called a non-convex set. A polygon that is not a convex polygon is sometimes called a concave polygon, and some
Convex_set
Function in mathematical analysis
Schur, Schur-convex functions are used in the study of majorization. A function f is 'Schur-concave' if its negative, −f, is Schur-convex. Every function
Schur-convex_function
Subfield of mathematical optimization
over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms
Convex_optimization
Simple polygon which is not convex
A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that
Concave_polygon
Mathematical function with convex lower level sets
quasiconvex function that is neither convex nor continuous. Convex function Concave function Logarithmically concave function Pseudoconvexity in the sense
Quasiconvex_function
Topics referred to by the same term
mirror Concave function, the negative of a convex function Concave polygon, a polygon which is not convex Concave set The concavity of a function, determined
Concave
Type of mathematical function
In convex analysis, a non-negative function f: Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies
Logarithmically concave function
Logarithmically_concave_function
Visual artifacts in ivory cross-sections
two forms: concave angles and convex angle. Concave angles have slightly concave sides and open to the medial (inner) area of the tusk. Convex angles have
Schreger_line
Gives conditions that guarantee the max–min inequality holds with equality
compact and convex, and to functions that are concave in their first argument and convex in their second argument (known as concave-convex functions).
Minimax_theorem
Algebra theorem about convex functions
for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes
Karamata's_inequality
Polyhedron with 12 faces
three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra
Dodecahedron
Curve in a liquid's surface due to adhesion to the container walls
water and glass. Water-based fluids like sap, honey, and milk also have a concave meniscus in glass or other wettable containers. Conversely, a convex meniscus
Meniscus_(liquid)
Polyhedron with some pattern of nonconvexity
which self-intersect in a repetitive way. Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way
Star_polyhedron
Tool used to measure the radius of an object
checked and any light leakage between the blade and edge indicates a mismatch that requires correction. A good set of gauges will offer both convex and concave
Radius_gauge
Set-to-real map with diminishing returns
very similar to convex and concave functions. For this reason, an optimization problem which concerns optimizing a convex or concave function can also
Submodular_set_function
Convex continuous functions on compact convex sets maximize at extreme points
analogue minimum principle: Any function that is concave and continuous, and defined on a set that is convex and compact, attains its minimum at some extreme
Bauer_maximum_principle
Four-sided polygon
or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees
Quadrilateral
convex-concave if it is a convex function of x for any fixed y, and a concave function of y for any fixed x. When f is continuous and convex-concave,
Min-max_optimization
Pentagon with all sides equal but the angles may not be equal
that don't intersect themselves are called simple, and they can be classified as either convex or concave. We here use the term "stellated" to refer to the
Equilateral_pentagon
Concept in convex analysis
particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain
Proper_convex_function
Quadrilateral symmetric across a diagonal
constructed from the centers and crossing points of any two intersecting circles. Kites as described here may be either convex or concave, although some sources
Kite_(geometry)
Theorem in geometry
arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex), then the area of the parallelogram is half the area
Varignon's_theorem
Lithograph print by M. C. Escher, first printed in 1957
characterized by shifting perception of depth from concave to convex depending on direction of light and shadow. Escher's interest in reversible perspectives
Cube_with_Magic_Ribbons
Toy lenses for Nikon SLR cameras
doublet, with either the convex or concave side facing the object. It is possible to thread the cemented doublet, with its concave side facing the object
Nikon_Amusing_Lenses
Shape with three sides
either convex (bending outward) or concave (bending inward). The intersection of three disks forms a circular triangle whose sides are all convex. An example
Triangle
Partition of a simple polygon into triangles
that keeps separate lists of convex and concave vertices will run in O(n2) time. This method is known as ear clipping and sometimes ear trimming. An efficient
Polygon_triangulation
Attraction that uses curved mirrors
carnivals and fairs. Instead of a normal plane mirror that reflects a perfect mirror image, distorting mirrors are curved mirrors, often using convex and concave
Distorting_mirror
Triangle with circular arc edges
mixture of convex and concave circular arc edges. Three given angles θ 1 {\displaystyle \theta _{1}} , θ 2 {\displaystyle \theta _{2}} , and θ 3 {\displaystyle
Circular_triangle
Formula for the area of a quadrilateral
expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not. The German mathematician
Bretschneider's_formula
Inequality between integrals in Lp spaces
another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named for
Hölder's_inequality
Smallest convex set containing a given set
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
Convex_hull
Polygon in which all angles are right
number of convex corners and Y the number of concave corners. By the previous fact, X=Y+4. Let XX the number of convex corners followed by a convex corner
Rectilinear_polygon
Extremely flat piece of optical-grade glass
indicate high and low points on the surface, such as rounded edges, hills or valleys, or convex and concave surfaces. Both the optical flat and the surface
Optical_flat
Optical aberration
lens systems, aberrations can be minimized using combinations of convex and concave lenses, or by using aspheric lenses or aplanatic lenses. Lens systems
Spherical_aberration
Engineering designation
dictionaries and encyclopedias mention male and female screws or cochleae. A 1736 builder's manual mentions screw genders as metaphors for convex and concave shapes:
Gender of connectors and fasteners
Gender_of_connectors_and_fasteners
Disproven hypothesis
sometimes called a "convex" Hollow Earth hypothesis, it is hypothesized humans live on the interior surface. This has been called the "concave" Hollow Earth
Hollow_Earth
Flat-sided three-dimensional shape
hosohedra) have no flat-faced analogue. If faces are allowed to be concave as well as convex, adjacent faces may be made to meet together with no gap. Some
Polyhedron
Architectural practice of cutting grooves through an otherwise plain surface
columns and decorative pilasters. Ancient Egyptian architecture used fluting in many buildings; most often the flutes are convex rather than concave, so the
Fluting_(architecture)
Point where the curvature of a curve changes sign
it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function
Inflection_point
Mathematical concept
front, convex or concave. Definition For a minimization problem with objective functions f 1 , … , f k {\displaystyle f_{1},\dots ,f_{k}} and the ideal
Multi-objective_optimization
Mathematical result in convex functions theory
is the convex conjugate of f {\displaystyle f} (also referred to as the Fenchel–Legendre transform) and g ∗ {\displaystyle g_{*}} is the concave conjugate
Fenchel's_duality_theorem
Optical illusion
staircase given to M. C. Escher by Prof. Schouten and which was an inspiration for Escher's Convex and Concave. This illusion is also seen in another Escher's
Schroeder_stairs
Polyhedron with 6 faces
hexahedron with all its faces square, and three squares around each vertex. There are seven topologically distinct convex hexahedra, one of which exists in
Hexahedron
S-curved form used in woodworking, moulding, textile weaving, and architecture
point of inflection from concave to convex or vice versa, have ends of the overall curve that point in opposite directions (and have tangents that are approximately
Ogee
fellow-workers and students had been as alert as he, they might even have invented the art of photography since al-Haytham's experiments with convex and concave mirrors
History_of_the_camera
Determining where a point is in relation to a coplanar polygon
The modified-polygon method, published in 2019, covers both convex and concave cases, and is based on the basic definition of the polygon size (line/area/volume)
Point_in_polygon
American mathematician
1A: Differentiation". Jerison, David. Eigenfunctions and harmonic functions in convex and concave domains. In: Srishti D. Chatterji (ed.): Proceedings
David_Jerison
Generalization of the normal-form game
actions, which is a simplex in Rmi. In a concave game, the set of strategies available to each player may be any convex set in Rmi. 2. In a normal-form game
Concave_game
Traditional attraction
and glass panes to parts of the maze they cannot yet get to. Sometimes the mirrors may be distorted because of different curves, convex, or concave in
House_of_mirrors
Swiss architect (born 1950)
renowned for its diamond-shaped glass façade with convex and concave panes, creating a dynamic and transparent structure that reflects the fashion brand's
Jacques_Herzog
April 2012. Nordenskiöld, Erland (1926). "Miroirs Convexes et Concaves en Amérique" [Convex and Concave Mirrors in the Americas]. Journal de la Société
Mirrors in Mesoamerican culture
Mirrors_in_Mesoamerican_culture
Horse breed from the Iberian Peninsula
convex profile. Ultra convex and concave profiles are discouraged in the breed and are penalized in breed shows. Necks are long and broad, running to well-defined
Andalusian_horse
Print by M. C. Escher
inhabitants casually strolling and one dining at an outdoor table. Yet all the figures are dressed in identical attire and have featureless bulb-shaped
Relativity_(M._C._Escher)
Body part used by insects to fly
respectively convex and concave, while the primitive Postcubitus and the first vannal have each an anterior convex branch and a posterior concave branch. The
Insect_wing
Church in England
projections running full length, and with flat raised fillets along each face. The capital gadrooned (convex and concave) raised acabi mouldings follow
St Andrew and St Mary's Church, Stoke Rochford
St_Andrew_and_St_Mary's_Church,_Stoke_Rochford
Arch curvature or posterior occlusal plane
across arch, and across median plane, curvature or posterior occlusal plane. Arc of the curve, which is concave for mandibular teeth and convex for maxillary
Curve_of_Wilson
Angle between a liquid–vapor interface and a solid surface
proportional to the mean curvature. Solving the above equation for both convex and concave surfaces yields: cos ( θ ∓ α ) = A + B cos α a ± C sin ( θ ∓
Contact_angle
Catalan solid with 60 faces
dodecahedron has three symmetry positions, two on vertices, and one on a midedge: A concave pentakis dodecahedron replaces the pentagonal faces of a dodecahedron
Pentakis_dodecahedron
Extinct genus of molluscs
rounded keel-like venters. Venters on inner whorls are truncated and broadly convex to concave. Immature forms bear a resemblance to Grypoceras. R.C. Moore
Gryponautilus
Polygon that is the boundary of a convex set
given vertex and the two adjacent vertices). Every non-degenerate triangle is strictly convex. Convex curve – Type of plane curve Concave polygon – Simple
Convex_polygon
Optical illusion
perception of a concave mask of a face appears as a normal convex face. While a convex face will appear to look in a single direction, and the gaze of a
Hollow-Face_illusion
Apeirohedron Regular skew polyhedron Abstract polytope 11-cell 57-cell Convex polygon Concave polygon Constructible polygon Cyclic polygon Equiangular polygon
List_of_mathematical_shapes
Ability of a liquid to maintain contact with a solid surface
to the mean curvature of the droplet, and is non zero. Solving the above equation for both convex and concave surfaces yields: cos ( θ ∓ α ) = A +
Wetting
Type of lens attached to a variety of optical devices such as telescopes and microscopes
add a simple positive, concave-convex lens before the doublet, with the concave face towards the light source and the convex surface facing the doublet
Eyepiece
Mechanical linkage consisting of four links connected by joints in a loop
quadrilateral linkage may be classified into three types: convex, concave, and crossing. In the convex and concave cases no two links cross over each other. In the
Four-bar_linkage
Convex molding in classical architecture
V-shapes enriching the surface of the concave ovolo in many early cases. The description of ovolo as the fundamental convex quarter-round element underlying
Ovolo
Unrealised project by Le Corbusier
specific culture and landscape. It comprised four main elements: an administration area by the water in two slab blocks, convex and concave apartment blocks
Ville_Radieuse
actuation on the membrane, a concave shape is achieved. This method is used in MEMS deformable mirrors to create convex and concave mirrors. Electrostatic-pneumatic
Electrostatic–pneumatic activation
Electrostatic–pneumatic_activation
a convolution of log-concave measures is log-concave. Convex measure, a generalisation of this concept Logarithmically concave function Prékopa, A. (1980)
Logarithmically concave measure
Logarithmically_concave_measure
Art and practice of creating images by recording light
fellow-workers and students had been as alert as he, they might even have invented the art of photography since al-Haitham's experiments with convex and concave mirrors
Photography
Type of synovial joint with concave, convex surfaces
reciprocally concave and convex. It is found in the thumb, the thorax, the middle ear, and the heel. In a saddle joint, one bone surface is concave while another
Saddle_joint
Association football club in Turin, Italy
The two "Golden Stars for Sport Excellence" were located above the convex and concave section of Juventus's emblem. During the 1980s, the club emblem was
Juventus_FC
Product of the principal curvatures of a surface
three-dimensional space at a point is the product of the two principal curvatures, κ1 and κ2, at the given point: K = κ 1 κ 2 . {\displaystyle K=\kappa _{1}\kappa
Gaussian_curvature
Polygon in which all sides have equal length
polygon does not need to be a convex polygon: it could be concave or even self-intersecting. All regular polygons and edge-transitive polygons are equilateral
Equilateral_polygon
Form of vision aid
published the first correct explanation as to why convex and concave lenses could correct presbyopia and myopia. Early frames for glasses consisted of two
Glasses
Compound lens
single convex and concave lens design. Alvan Graham Clark and Bausch & Lomb further refined the design in 1888 by taking two of these lenses and placing
Double-Gauss_lens
Quadrilateral with two pairs of parallel sides
parallelogram, called its Varignon parallelogram. If the quadrilateral is convex or concave (that is, not self-intersecting), then the area of the Varignon parallelogram
Parallelogram
1996 video game
patterns by bending the lines or altering the colors, and can save and print the images. Convex and Concave is a game in which players are presented with perspective
Escher_Interactive
Method to solve optimization problems
principle for convex functions (alternatively, by the minimum principle for concave functions) since linear functions are both convex and concave. However
Linear_programming
Monument in Paarl, Western Cape, South Africa
Afrikaners' identity and pride in their language. The monument consists of various tapering structures of a convex and concave nature, symbolising the
Afrikaans_Language_Monument
Standards used to define surface quality
optics, but recently also for convex and concave lenses. In contrast, 'Roughness' characterization is done with more precise and easier-to-quantify methods
Surface imperfections (optics)
Surface_imperfections_(optics)
Optical illusion
the convex corner of a room, which is closer, and the "angles out" configuration corresponds to an object which is far away, such as the concave corner
Müller-Lyer_illusion
Dutch geometer and astronomer
of a convex and concave lens in a tube, and the combination magnified three or four times. Adriaan Metius attended a Latin school in Alkmaar and studied
Adriaan_Metius
Partial order on matrices
the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar
Loewner_order
Polytope or tiling with one type of edge
180} o , {\displaystyle {\color {royalblue}^{\mathsf {o}}},} making convex or concave polygons respectively. A star 2 n {\displaystyle {\color {royalblue}{\mathbf
Isotoxal_figure
Swiss architect (born 1950)
this building is recognized for its diamond-shaped glass façade with convex and concave panes. Tate Modern (2000) – The transformation of a disused power
Pierre_de_Meuron
Humanist sans-serif typeface
Calibri, Cambria, Consolas, Corbel and Constantia. Candara's verticals show both convex and concave curvature with entasis and ectasis on opposite sides of
Candara
Theorem of convex functions
opposite inequality for concave transformations). Jensen's inequality generalizes the statement that the secant line of a convex function lies above the
Jensen's_inequality
converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It
Concavification
Sculpture in Zagreb, Croatia
it is placed. The transitions between light and shadow are mild. The sculpture is both convex and concave.[citation needed] Auguste Rodin's influence
Well_of_Life_(sculpture)
Type of lens
of prototypes and small series. Like other lenses for vision correction, aspheric lenses can be categorized as convex or concave. Convex aspheric curvatures
Aspheric_lens
Species of plant
millimeters wide. Their upper side is flat to convex, but concave towards the base. The underside of the leaf is convex. The broadly egg-shaped, bare, gray-green
Crassula_rupestris
Property of a planar simple closed curve
sequence F-G-H is concave. The following table illustrates rules for determining whether a sequence of points is convex, concave, or flat: Convex hull Differential
Curve_orientation
Japanese luxury vehicle brand owned by Toyota
characteristics, including a fastback profile, lower-set grille, and the use of both convex and concave surfaces, are derived from Japanese cultural motifs (e.g
Lexus
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CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
CONVEX AND-CONCAVE
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