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Convex

In geometry, a convex set is a subset of an affine space that contains the line segment between any two points in the set. The concept of convexity extends to various fields, including mathematics (such as convex functions and optimization), physics and optics (like convex lenses and mirrors), economics and finance (convexity in preferences and financial instruments), and other applications in biology, anatomy, architecture, and design.

Mathematics

Convex Sets

In a real vector space, a set C is convex if, for any x, y \in C and any \theta \in [0, 1], the point \theta x + (1 - \theta) y also belongs to C.^[1] This condition ensures that the line segment connecting any two points in the set lies entirely within the set.^[1] Convex sets are closed under convex combinations, meaning that any finite weighted sum \sum_{i=1}^k \theta_i x_i, where x_i \in C, \theta_i \geq 0, and \sum_{i=1}^k \theta_i = 1, remains in C.^[1] An extreme point of a convex set C is a point x \in C that cannot be expressed as a nontrivial convex combination of other points in C.^[1] A supporting hyperplane to C at a boundary point x_0 \in C is defined by a nonzero vector a such that a^T x \leq a^T x_0 for all x \in C, with the hyperplane \{x \mid a^T x = a^T x_0\} touching C at x_0.^[1] Separation theorems provide a way to distinguish disjoint convex sets. For two nonempty disjoint convex sets C and D in a real vector space, where one is compact and the other closed, there exists a separating hyperplane: a nonzero a and scalar b such that a^T x \leq b for all x \in C and a^T x \geq b for all x \in D.^[1] This result follows from applications of the Hahn-Banach theorem, which extends linear functionals while preserving bounds, enabling strict separation under additional conditions like compactness.^[1] Common examples of convex sets include Euclidean balls \{x \mid \|x - x_c\|_2 \leq r\}, half-spaces \{x \mid a^T x \leq b\}, polyhedra formed as intersections of finitely many half-spaces (possibly with equality constraints), and simplexes, which are the convex hulls of n+1 affinely independent points in \mathbb{R}^n.^[1] The convex hull of a set S, denoted \operatorname{conv} S, is the smallest convex set containing S, consisting of all convex combinations of points from S.^[1] The Krein-Milman theorem states that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.^[2] This representation highlights the role of extreme points in characterizing the geometry of such sets.^[2] Convex sets form the domains for convex functions and the feasible regions in convex optimization problems.^[1]

Convex Functions

A convex function is defined on a convex set S \subseteq \mathbb{R}^n as a function f: S \to \mathbb{R} satisfying f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda) f(y) for all x, y \in S and \lambda \in [0,1].^[3] Equivalently, f is convex if its epigraph \{(x, t) \in S \times \mathbb{R} \mid f(x) \leq t\} is a convex set.^[4] The function is strictly convex if the inequality is strict for \lambda \in (0,1) and x \neq y.^[3] Key properties of convex functions include Jensen's inequality, which states that for a convex f and random variable z with support in \dom f, f(\mathbb{E}) \leq \mathbb{E}[f(z)].^[3] Convex functions are characterized by their subdifferentials: the subdifferential \partial f(x) at x \in \dom f is the set \{ g \in \mathbb{R}^n \mid f(y) \geq f(x) + g^T (y - x), \ \forall y \in \dom f \}, which is nonempty on the relative interior of the domain and generalizes the gradient for nondifferentiable cases.^[4] The Fenchel conjugate, or convex conjugate, of f is f^*(y) = \sup_{x \in \dom f} (y^T x - f(x)), which is always convex and lower semicontinuous, and satisfies f^{**} = f for proper, convex, lower semicontinuous f.^[3]^[4] Examples of convex functions include norms such as the Euclidean norm \|x\|_2 = \sqrt{\sum_i x_i^2}, which satisfy the triangle inequality and homogeneity.^[3] Quadratic functions f(x) = x^T P x with positive semidefinite P are convex, and the simple case f(x) = x^2 on \mathbb{R} illustrates strict convexity for positive definite P.^[3] The exponential function f(x) = e^{a^T x + b} is convex, as its second derivative is positive.^[3] Relatedly, log-concave functions arise inversely, as the logarithm of a positive log-concave function is concave, contrasting with the convexity of the exponential.^[3] Convexity is preserved under certain compositions: if h is convex and nondecreasing, and g is convex, then h(g(x)) is convex; more generally, for f(x) = h(g(x)), convexity holds if h is convex and each component of g is convex with h nondecreasing in that argument, or concave with h nonincreasing.^[3] Convex functions relate to quasiconvex functions, where sublevel sets \{x \mid f(x) \leq \alpha\} are convex for all \alpha; every convex function is quasiconvex, but not conversely, as quasiconvexity requires only f(\lambda x + (1-\lambda)y) \leq \max\{f(x), f(y)\}.^[3] These properties ensure that local minima of convex functions are global, facilitating their use in optimization over convex domains.^[4]

Convex Optimization

Convex optimization addresses the minimization of a convex objective function subject to convex constraints, a class of problems that admits efficient numerical solution methods and strong theoretical guarantees. The canonical formulation is to solve \min_{x} f(x) subject to x \in \mathcal{C}, where f is a convex function and \mathcal{C} is a convex set defining the feasible region.^[5] Common standard forms include linear programming (\min c^\top x subject to Ax \leq b, x \geq 0), quadratic programming (with quadratic objectives like \frac{1}{2} x^\top Q x + c^\top x where Q \succeq 0), and semidefinite programming (minimizing linear functions over spectrahedra defined by linear matrix inequalities).^[5] These forms encompass a wide range of practical problems due to their tractability and the fact that local minima coincide with global minima.^[6] Key theoretical results underpin the reliability of convex optimization. For unconstrained problems with strictly convex objectives, a minimizer exists and is unique if the function is coercive.^[5] In constrained settings, the Karush-Kuhn-Tucker (KKT) conditions provide necessary and sufficient optimality criteria under convexity, stating that at an optimum x^*, there exist multipliers \lambda^* such that the stationarity condition \nabla f(x^*) + \sum \lambda_i^* \nabla g_i(x^*) = 0 holds, alongside primal feasibility, dual feasibility, and complementary slackness.^[7] Strong duality, ensuring zero duality gap between primal and dual problems, follows from Slater's condition: for convex problems with inequality constraints, if there exists a strictly feasible point in the relative interior of the feasible set, the dual attains the primal optimum.^[5] Several algorithms exploit convexity for efficient solutions, often achieving polynomial-time complexity. The ellipsoid method, introduced by Khachiyan in 1979, was the first to demonstrate polynomial-time solvability for linear programs by iteratively shrinking ellipsoids containing an optimum using separating hyperplanes.^[8] Interior-point methods, pioneered by Karmarkar in 1984 for linear programming, follow central paths via barrier functions and extend to general convex problems with self-concordant barriers, achieving O(\sqrt{n} \log(1/\epsilon)) iterations for \epsilon-accuracy in n dimensions.^[9] First-order methods like gradient descent converge at O(1/k) for smooth convex functions after k iterations, while Nesterov's accelerated variant from 1983 improves this to O(1/k^2) by incorporating momentum terms.^[10] Applications of convex optimization span machine learning and signal processing, with historical roots in operations research. In machine learning, support vector machines reduce to convex quadratic programs for maximum-margin classification, as formulated by Cortes and Vapnik in 1995. In signal processing, convex formulations enable sparse recovery via basis pursuit and robust beamforming.^[5] The field's modern development accelerated in the 1980s with Nesterov's optimal methods for smooth convex problems, influencing subsequent algorithmic advances.^[10]

Physics and Optics

Convex Lenses

A convex lens is a transparent optical element with at least one surface bulging outward (convex), typically thicker at the center than at the edges, causing parallel rays of light to converge to a real focal point after refraction.^[11]^[12] The key properties of convex lenses include their converging nature, which results in the formation of real, inverted, and magnified images for objects placed beyond the focal point, while objects within the focal length produce virtual, upright, and magnified images.^[13] The focal length f of a thin convex lens is determined by the lensmaker's formula: \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right), where n is the refractive index of the lens material, and R_1 and R_2 are the radii of curvature of the lens surfaces (positive for convex surfaces facing the incident light from the left).^[11]^[14] For a symmetric biconvex lens, R_1 = -R_2 = R > 0, simplifying to \frac{1}{f} = 2(n - 1)/R. In the standard sign convention for lenses, f is positive for converging lenses.^[13] Image formation in convex lenses follows the thin lens equation \frac{1}{o} + \frac{1}{i} = \frac{1}{f}, where o is the object distance (positive for real objects to the left of the lens), i is the image distance (positive for real images to the right, negative for virtual images to the left), and f is the focal length (positive for convex lenses).^[13] The magnification m = -\frac{i}{o} is negative for real inverted images (with |m| > 1 for magnification when o > f) and positive for virtual upright images (with |m| > 1 when o < f).^[15] Practical applications of convex lenses exploit their ability to focus light, such as in eyeglasses for correcting farsightedness (hyperopia), cameras and projectors to form sharp images on sensors or screens, and microscopes or telescopes as objective lenses to magnify distant or small objects.^[16] They are also used in magnifying glasses for close-up viewing, where the lens held close to the eye creates a virtual enlarged image. These converging properties make convex lenses fundamental for imaging systems requiring magnification or focusing, contrasting with the diverging role of convex mirrors.^[12]

Convex Mirrors

A convex mirror is a spherical mirror with its reflective surface bulging outward toward the incident light, causing parallel rays to diverge after reflection as if emanating from a virtual focal point located behind the mirror.^[17]^[18] The key properties of convex mirrors include their diverging nature, which results in the formation of virtual, upright, and diminished images for all object positions, with the image appearing smaller than the object and located behind the mirror.^[17] Unlike plane mirrors, convex mirrors provide a wider field of view, allowing observation of a broader area without moving the observer.^[18]^[19] The focal length f of a convex mirror is related to its radius of curvature R by f = \frac{R}{2}, where f and R are negative in the standard sign convention for mirrors, indicating the virtual focus behind the reflecting surface.^[17]^[20] Image formation in convex mirrors is described by the mirror equation \frac{1}{o} + \frac{1}{i} = \frac{1}{f}, where o is the object distance (positive for real objects in front of the mirror), i is the image distance (negative for virtual images behind the mirror), and f is the focal length (negative for convex mirrors).^[17] This equation, analogous to that for lenses but applied to reflection, ensures that the image is always virtual and reduced in size, with magnification m = -\frac{i}{o} yielding a positive value less than 1, confirming the upright and diminished characteristics.^[17] Practical applications of convex mirrors leverage their wide-angle viewing capability and safety benefits, such as in passenger-side rear-view mirrors in vehicles, which reduce blind spots by expanding the visible field behind the driver compared to flat mirrors.^[18]^[19] They are also commonly used as security mirrors in stores and hallways to monitor larger areas for surveillance and prevent accidents by providing a panoramic view around corners.^[18]^[21] These diverging properties make convex mirrors essential for applications requiring broad visibility rather than image magnification, complementing the focusing role of convex lenses in optical systems.^[17]

Economics and Finance

Convexity in Finance

In finance, convexity measures the curvature in the relationship between a bond's price and its yield to maturity, serving as a second-order sensitivity metric beyond duration.^[22] Mathematically, it is defined as the second derivative of the bond price P with respect to the yield y, scaled by the price: C = \frac{1}{P} \frac{d^2 P}{dy^2}. For practical computation, convexity approximates as C \approx \frac{1}{P} \sum_{t=1}^T \frac{t(t+1) CF_t}{(1+y)^{t+2}}, where CF_t represents the cash flow at time t and T is the bond's maturity.^[23] This formula captures how bond prices respond non-linearly to yield changes, with the price-yield curve exhibiting convexity akin to the mathematical property of convex functions.^[22] For standard fixed-income securities like non-callable bonds, convexity is positive, meaning the bond's price rises more when yields fall than it falls when yields rise by the same amount, thereby cushioning downside risk.^[24] This property enhances the first-order approximation provided by duration, which measures linear price sensitivity (\frac{dP}{dy}); convexity refines this by accounting for the curve's bend, improving accuracy for larger yield shifts.^[23] In portfolio contexts, bonds with higher convexity offer greater protection against adverse interest rate movements, such as parallel shifts in the yield curve, as the positive curvature amplifies gains relative to losses.^[24] Convexity finds key applications in portfolio immunization, where matching both duration and convexity between assets and liabilities minimizes interest rate risk under non-parallel yield curve changes. It also plays a role in calculating option-adjusted spreads (OAS) for bonds with embedded options, such as mortgage-backed securities, by adjusting for the convexity effects of prepayment or call features that introduce negative convexity.^[25] Higher convexity in a portfolio reduces overall risk exposure to yield curve shifts, enabling better hedging and risk-adjusted returns.^[26] The concept of convexity emerged in fixed-income theory during the 1970s, building on duration measures amid rising interest rate volatility, with early formalization attributed to economists like Hon-Fei Lai and popularization by Stanley Diller at firms such as Goldman Sachs. It provided a superior approximation to duration for bond pricing and risk management, addressing limitations in linear models during that era's market turbulence.^[27]

Convex Preferences

In microeconomics, convex preferences describe a consumer's ordering of consumption bundles such that the set of bundles at least as preferred as any given bundle forms a convex set. Formally, a preference relation \succsim on a convex consumption set X \subseteq \mathbb{R}^l_+ is convex if, for every y \in X, the upper contour set \{x \in X \mid x \succsim y\} is convex.^[28] This property captures the intuitive notion that consumers prefer diversified bundles, often visualized as indifference curves that bow toward the origin. When preferences admit a utility representation u: X \to \mathbb{R}, convexity is equivalent to u being quasiconcave, meaning that for any x, y \in X with u(x) = u(y), the utility along any convex combination \lambda x + (1-\lambda) y (for \lambda \in [0,1]) is at least as high as u(x). A key property of convex preferences is the diminishing marginal rate of substitution (MRS), where the amount of one good a consumer is willing to forgo for an additional unit of another good decreases as the quantity of the first good rises; this ensures smoother substitution possibilities and aligns with observed consumer behavior. Additionally, convexity facilitates the existence of Walrasian equilibria in general equilibrium models, as it guarantees that aggregate excess demand functions satisfy the necessary continuity and boundary conditions for equilibrium prices.^[29] Representative examples include the Cobb-Douglas utility function u(x_1, x_2) = x_1^a x_2^{1-a} for $0 < a < 1, which is strictly quasiconcave and thus represents strictly convex preferences, leading to indifference curves with a constant-elasticity MRS.^[30] In settings with uncertainty, such as lotteries over outcomes, convex preferences arise under expected utility theory when the von Neumann-Morgenstern utility function is concave, implying risk-averse behavior where the consumer prefers a sure payoff to a fair gamble. Seminal results include Debreu's representation theorem, which establishes that continuous preferences on a connected, convex consumption set can be represented by a continuous utility function, with convexity ensuring the representing utility is quasiconcave.^[31] Furthermore, strict convexity of preferences—where strict upper contour sets are convex and mixtures are strictly preferred—implies unique optimal consumption bundles for given budget sets, eliminating multiple demand solutions and strengthening equilibrium uniqueness properties.

Other Uses

Convex Shapes in Biology and Anatomy

In biological structures, convex shapes often confer functional advantages by distributing mechanical stresses, facilitating fluid dynamics, or enhancing optical performance. These forms arise through evolutionary adaptations, optimizing survival in diverse environments. For instance, the convex curvature of the human spine, particularly in the lumbar region (lordosis), enables efficient load-bearing and shock absorption during movement. Intervertebral discs, positioned between vertebrae, adopt a wedge-like convexity that contributes to this overall curve, allowing the spine to flex while maintaining structural stability. This design dissipates axial forces from activities like walking or jumping, reducing injury risk to the spinal cord and nerves.^[32]^[33] In ocular anatomy, convex forms are critical for refraction. The human cornea, with its convex anterior surface, accounts for about two-thirds of the eye's total refractive power, bending incoming light rays to focus them on the retina.^[34] This curvature, approximately 7.8 mm in radius, ensures clear vision by compensating for the lower refractive index of air relative to the corneal tissue. Early anatomical descriptions, such as those by Andreas Vesalius in his 1543 work De humani corporis fabrica, illustrated the eye's layered structure.^[35] In aquatic animals, similar principles apply but are adapted for underwater optics. Fish eyes feature a nearly spherical lens that is highly convex, providing the strong refraction needed in water's higher refractive index to focus images on the retina without relying on the cornea, which loses effectiveness when immersed. This adaptation allows species like salmon to achieve emmetropia (focused vision) across vast underwater distances.^[36]^[37] Convexity also appears in plant morphology for environmental resilience. Many tropical leaves exhibit convex drip tips or overall upward curvature, which accelerates water shedding during heavy rainfall, preventing fungal growth and optimizing photosynthesis by minimizing prolonged wetness. This shape reduces droplet retention time by up to 50% compared to flat leaves, as demonstrated in studies of leaf apex structures where convex forms promote rapid drainage via gravity and surface tension. In evolutionary terms, such adaptations likely emerged in humid ecosystems to enhance pathogen resistance and nutrient uptake efficiency.^[38] Among birds, shorebirds like curlews and ibises possess long, gently convex (downward-curving) bills that enable probing into soft substrates such as mud or soil to extract invertebrates, a trait refined through natural selection for efficient prey capture in wetland habitats. This curvature increases mechanical leverage and reach without compromising structural strength, allowing repeated insertions with minimal energy expenditure. Evolutionary analyses of beak morphometry across avian lineages show that such convex forms correlate with probing behaviors, diverging from straighter bills used for seed-cracking or nectar-sipping.^[39]^[40] Skeletal elements in mammals further illustrate convex adaptations for integrity. The scapula, or shoulder blade, features a slightly convex posterior surface in many species, which enhances muscle attachment and load distribution during locomotion. This form evolved from early therian ancestors, where phylogenetic shifts in scapular shape supported terrestrial gaits, providing resilience against torsional forces in quadrupeds and primates alike. In pathological contexts, abnormal convex spinal curvatures manifest as conditions like thoracic scoliosis, where the spine's lateral convexity exceeds 10 degrees, potentially compressing organs and altering organ anatomy due to asymmetric growth. Studies link scoliosis convexity direction (rightward in 80-99% of cases) to underlying vascular or neural asymmetries, influencing treatment approaches.^[41]^[42]

Convex Forms in Architecture and Design

Convex forms, characterized by their outward-bulging shapes, have been integral to architectural design for their ability to enhance structural integrity and create dynamic visual effects. In Baroque architecture of the 17th century, architects employed convex elements to introduce movement and drama, replacing the straight lines of Renaissance designs with flowing curves that alternated between projections and recesses. For instance, facades often featured undulating convex surfaces to play with light and shadow, fostering a sense of theatricality and spatial depth. Domes, too, incorporated convex profiles to symbolize grandeur and illusionistic expansion, as seen in structures like St. Peter’s Basilica in Rome, where curved forms contributed to the style's emphasis on opulence and power.^[43] A notable early application of convex forms appears in historical art integrated into architecture, such as the convex mirror in Jan van Eyck's 1434 painting The Arnolfini Portrait. This mirror, positioned on the back wall of a depicted domestic interior, reflects the room's occupants and two entering figures, demonstrating advanced optical realism and serving as a symbolic element that expands the viewer's perception of space. In structural engineering, convex arches in bridges exemplify practical benefits, where the upward-curving form transfers vertical loads into horizontal thrust, distributing weight efficiently to abutments and minimizing bending moments and shear forces in the structure. This design reduces material stress, allowing for spans with less material while maintaining stability, as utilized in ancient Roman bridges and persisting in modern iterations.^[44]^[45] In contemporary architecture, convex facades have gained prominence through parametric design tools like Rhinoceros (Rhino) software, developed in the post-2000s era to model complex curves algorithmically. The Heydar Aliyev Center in Baku, Azerbaijan, designed by Zaha Hadid Architects and completed in 2012, features fluid convex surfaces that bulge outward in seamless, organic waves, creating an illusion of continuous movement and integrating the building with its landscape. Similarly, convex glass elements in windows and facades, often using low-iron glass with Low-E coatings, enhance aesthetic flow while providing functional advantages; for example, the undulating convex panels at 2100 Pennsylvania Avenue in Washington, DC, produce a "waving flag" visual effect that unifies the structure's form. These shapes also offer engineering merits, such as improved wind resistance, where convex configurations promote smoother airflow and reduce stagnation zones around building clusters by up to 20% in simulated models, thereby enhancing overall stability against environmental loads.^[46]^[47]^[48]^[49]

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