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Mathematical object
In mathematics, a random polytope is a structure commonly used in convex analysis and the analysis of linear programs in d-dimensional Euclidean space
Random_polytope
Fundamental theorem in probability theory and statistics
than 2. The polytope Kn is called a Gaussian random polytope. A similar result holds for the number of vertices (of the Gaussian polytope), the number
Central_limit_theorem
needed] 142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated
List_of_mathematical_shapes
In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial that can be used in the asymptotic analysis of those
Newton_polytope
Type of probability distribution
Tomczak-Jaegermann, N. (2005). "Smallest singular value of random matrices and geometry of random polytopes" (PDF). Advances in Mathematics. 195 (2): 491–523.
Sub-Gaussian_distribution
Polytope
The Birkhoff polytope B n {\displaystyle B_{n}} is the convex polytope in R n 2 {\displaystyle \mathbb {R} ^{n^{2}}} whose points are the doubly stochastic
Birkhoff_polytope
mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic
Order_polytope
economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the
Stable_matching_polytope
Multi-dimensional generalization of triangle
dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point
Simplex
Natural number
zonotope. Seventeen is the highest dimension for paracompact Vineberg polytopes with rank n + 2 {\displaystyle n+2} mirror facets, with the lowest belonging
17_(number)
Shape where all small sets of vertices form a face
k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which
Neighborly_polytope
is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly
Unique_sink_orientation
x m } {\displaystyle K_{m}={\text{conv}}\{x_{1},...,x_{m}\}} is a random polytope. Intuitively, it is clear that as m → ∞ {\displaystyle m\rightarrow
Convex_cap
Algorithm for linear programming
neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function
Simplex_algorithm
Greek-French composer, architect and engineer (1922–2001)
Xenakis's UPIC system; and the massive multimedia performances Xenakis called polytopes, that were a summa of his interests and skills. Among the numerous theoretical
Iannis_Xenakis
Method to solve optimization problems
affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or
Linear_programming
Shape with four equal sides and angles
truncated square is an octagon. The square belongs to a family of regular polytopes that includes the cube in three dimensions and the hypercubes in higher
Square
Branch of discrete mathematics
convex polytope can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special
Combinatorics
Branch of geometry that studies combinatorial properties and constructive methods
a polyhedron or polytope, unit disk graphs, and visibility graphs. Topics in this area include: Graph drawing Polyhedral graphs Random geometric graphs
Discrete_geometry
Geometric space with four dimensions
both synthetic and algebraic methods. He discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean
Four-dimensional_space
stabilization state feedback synthesis for systems characterized by random polytopes". 2016 European Control Conference (ECC). pp. 2023–2028. doi:10.1109/ecc
Finsler's_lemma
Unit hypercube of variable dimension whose corners have been perturbed
The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty) is a unit hypercube of variable dimension whose corners have been
Klee–Minty_cube
Award for advancements in discrete mathematics
characterization of the weakly bipartite graphs (graphs whose bipartite subgraph polytope is 0-1). Satoru Iwata, Lisa Fleischer, Satoru Fujishige, and Alexander
Fulkerson_Prize
Business Media. pp. 15-16. ISBN 0-387-97993-X. "Infinity Scrapers". www.polytope.net. Retrieved 7 December 2025. "How we started and where we are today"
Names_of_large_numbers
Graph theory model
k-dimensional stacked polytopes, polytopes formed by starting from a simplex and then repeatedly gluing simplices onto the faces of the polytope, are k-trees when
K-tree
Concept in integral mathematics
programming on this polytope would automatically yield the correct solution to the original integer program. However, in general, this polytope will have exponentially
Linear_programming_relaxation
Manifold of dimension 3 equipped with a hyperbolic metric
Coxeter polytopes (polytopes whose dihedral angles are of the form π / m , m ∈ N {\displaystyle \pi /m,m\in \mathbb {N} } ). Such a polytope gives rise
Hyperbolic_3-manifold
different blockchains. Hyperbridge was launched on Polkadot mainnet in 2024 by Polytope Labs. The protocol is designed with a decentralised verification system
Hyperbridge
Canadian geometer (1907–2003)
author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him
Harold Scott MacDonald Coxeter
Harold_Scott_MacDonald_Coxeter
Type of probability distribution
distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated
Truncated_normal_distribution
Generalized sphere of dimension n (mathematics)
^{n+1}:\left\|x\right\|_{1}=1\right\}} In general, it takes the shape of a cross-polytope. The octahedral 1 {\displaystyle 1} -sphere is a square (without its
N-sphere
Provides lower bounds on the circuit complexity of boolean functions
or "variation" of the values of a Boolean function, or of an associated polytope or other structure. [...] Show by an inductive argument that polynomial-sized
Natural_proof
Number associated with symmetric convex bodies
polyhedron or polytope is its dual polyhedron or dual polytope. In particular, the polar body of a cube or hypercube is an octahedron or cross polytope. Its Mahler
Mahler_volume
Set of vectors used to define coordinates
general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis consists of
Basis_(linear_algebra)
Partition of a simple polygon into triangles
create a triangulation based on a set of points. The associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon. Polygon
Polygon_triangulation
Area of discrete mathematics
Exploring Structure and Randomness. Cambridge University Press. ISBN 978-1-009-31094-9. Ziegler, Günter M. (2007). "Convex polytopes: extremal constructions
Graph_theory
Tool for working with matrices
applications. One such application is for the problem of fair random assignment: given a randomized allocation of items, Birkhoff's algorithm can decompose
Birkhoff_algorithm
Graph formed by subdivision of triangles
graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing
Apollonian_network
Complex structures in matter physics
regular tetrahedra if the space is not Euclidean, but spherical. It is the polytope {3,3,5}, using the Schläfli notation, also known as the 600-cell. There
Geometrical_frustration
Sequence of operations for a task
the volume of a convex polytope (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm
Algorithm
Topics referred to by the same term
polyhedron, a generalization of a regular polygon to higher dimensions Regular polytope, a generalization of a regular polygon to higher dimensions Regular skew
Regular
Type of cellular automaton
three-dimensional grid, the only regular polytope that fills the whole space is the cube, while the only regular polytopes with a sufficiently large symmetry
Lattice_gas_automaton
in dimension 4 not projectively equivalent to the vertices of a convex polytope", Combinatorial geometries (Luminy, 1999), European Journal of Combinatorics
Convex_position
Deviations from local realism
In that representation, the set of all classical boxes forms a convex polytope. In the Bell scenario studied by CHSH, where a , b , x , y {\displaystyle
Quantum_nonlocality
NP-hard problem in combinatorial optimization
Mark (2017). "Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem". Operations
Travelling_salesman_problem
Covering by shapes without overlaps or gaps
pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more
Tessellation
Prime number of the form 2^n – 1
geometry, the number of polytopes that are part of the family of polytopes formed by a truncation operation of a base regular polytope and its dual (excluding
Mersenne_prime
All numbers between two given numbers
of half-spaces (of arbitrary orientation) is (the interior of) a convex polytope, or in the 2-dimensional case a convex polygon. An open interval is a connected
Interval_(mathematics)
of objects is induced by dilations and translations of a fixed convex polytope. He proved upper and lower bounds on the discrepancy. The results are analogous
Geometric_discrepancy
Study of graphs defined by geometric means
points. The 1-skeleton of a polyhedron or polytope is the set of vertices and edges of said polyhedron or polytope. The skeleton of any convex polyhedron
Geometric_graph_theory
Triangulation method
29 October 2018. Seidel, Raimund (1995). "The upper bound theorem for polytopes: an easy proof of its asymptotic version". Computational Geometry. 5 (2):
Delaunay_triangulation
Study of mathematical algorithms for optimization problems
equalities and inequalities. Such a constraint set is called a polyhedron or a polytope if it is bounded. Second-order cone programming (SOCP) is a convex program
Mathematical_optimization
Type of plane partition
points and all of them are different, then the Voronoi cells are convex polytopes and they can be represented in a combinatorial way using their vertices
Voronoi_diagram
Class of algorithms in computational geometry
hull is a convex polyhedron in three dimensions, or in general a convex polytope for any number of dimensions, whose vertices are some of the points in
Convex_hull_algorithms
parallelohedron? Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram? Ropelength
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
rather than by an explicit listing of the vertices or faces of a convex polytope. It is known that, in this model, no deterministic algorithm can achieve
Convex_volume_approximation
Triangular array of the binomial coefficients
(1973-01-01). "Chapter VII: ordinary polytopes in higher space, 7.2: Pyramids, dipyramids and prisms". Regular Polytopes (3rd ed.). Courier Corporation. pp
Pascal's_triangle
Smallest convex set containing a given set
Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to S {\displaystyle
Convex_hull
German mathematician and physicist (1864–1909)
Minkowski model Minkowski plane Minkowski problem Minkowski problem for polytopes Minkowski's second theorem Minkowski space Minkowski's bound Minkowski's
Hermann_Minkowski
Computer science award
doi:10.1145/2716307. S2CID 7372000. Rothvoss, Thomas (2017). "The Matching Polytope has Exponential Extension Complexity". Journal of the ACM. 64 (6): 41:1–41:19
Gödel_Prize
Basic concept of graph theory
states that the polytopal graph (1-skeleton) of a k-dimensional convex polytope is a k-vertex-connected graph. Steinitz's previous theorem that any 3-vertex-connected
Connectivity_(graph_theory)
shapes List of matrices List of numbers List of polygons, polyhedra and polytopes List of prime numbers —not merely a numerical table, but a list of various
List_of_mathematical_examples
Advanced method of process control
of all the regions. Every region turns out to geometrically be a convex polytope for linear MPC, commonly parameterized by coefficients for its faces, requiring
Model_predictive_control
Natural number
three-dimensional uniform polyhedra that are cell facets inside uniform 4-polytopes that are not part of infinite families of antiprismatic prisms and duoprisms:
23_(number)
doi:10.1088/1475-7516/2007/01/004. S2CID 17403084. "Infinity Scrapers". www.polytope.net. Retrieved 2025-12-07. "Forcal - Aarex's Large Numbers". sites.google
Orders_of_magnitude_(numbers)
Group of rotations in 3 dimensions
^{n}} expressed in its standard basis. Coxeter, H. S. M. (1973). Regular polytopes (Third ed.). New York: Dover Publications, Inc. p. 53. ISBN 0-486-61480-8
3D_rotation_group
Concept in geometry
resort to "throwing darts". This Monte Carlo method uses the fact that if random samples are taken uniformly scattered across the surface of a square in
Area_of_a_circle
Topics referred to by the same term
compound Common Random Numbers, a statistical procedure Concentration ratio, a measure of market concentration in economics Cross-polytope of n-dimensions
CRN
American operations manager
Lifting the Facets of O-1 Polytopes. Vol. 15. Mathematical Programming. pp. 268–277. Zemel, E. (1987). A Linear Time Randomizing Algorithm for Searching
Eitan_Zemel
Infinitely detailed mathematical structure
two quantities Publications in fractal geometry Random walk – Process forming a path from many random steps Self-reference – Sentence, idea or formula
Fractal
Periodic set of points
S} . For a polytope whose vertices are elements of the lattice, the number of lattice points it contains is described by the polytope's Ehrhart polynomial
Lattice_(group)
Pairing where no unchosen pair prefers each other over their choice
matching problems Rainbow matching for edge colored graphs Stable matching polytope Secretary problem (also called marriage problem) – deciding when to stop
Stable_matching_problem
Matrix with exactly one 1 per row and column
stochastic matrices is called the Birkhoff polytope, and the permutation matrices play a special role in that polytope. The Birkhoff–von Neumann theorem says
Permutation_matrix
On tangency patterns of circles
5935240, ISBN 978-1-4244-9919-9 Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, pp. 117–118
Circle_packing_theorem
Combinatorial algorithm
Equivalently, this algorithm finds a Hamiltonian cycle in the permutohedron, a polytope whose vertices represent permutations and whose edges represent swaps.
Steinhaus–Johnson–Trotter algorithm
Steinhaus–Johnson–Trotter_algorithm
Interplay between observation, experiment, and theory in science
mathematicians, of Euler's formula for polyhedra. H.S.M. Coxeter (1973) Regular Polytopes ISBN 9780486614809, Chapter IX "Poincaré's proof of Euler's formula" Charles
Scientific_method
Idea that the universe is a digital computation device
Datenverarbeitung vol 8., pages 336–344 Fritz, Tobias (June 2013). "Velocity polytopes of periodic graphs and a no-go theorem for digital physics". Discrete
Digital_physics
Czech-Canadian mathematician
Mathematics of Operations Research, 1979 Chvátal, Václav (1973), "Edmonds polytopes and weakly hamiltonian graphs", Mathematical Programming, 5: 29–40, doi:10
Václav_Chvátal
Estimation approach for random vectors by representing them as sets
the corresponding feasible set for the state vector can be described by polytopes or by ellipsoids . When the system is nonlinear, the set can be enclosed
Set_estimation
Invariant measure of fractal dimension
of fractals by Hausdorff dimension Examples of deterministic fractals, random and natural fractals. Assouad dimension, another variation of fractal dimension
Hausdorff_dimension
Statistical method
square rotated so that its corners lie on the axes (in general a cross-polytope), while the region defined by the ℓ 2 {\displaystyle \ell ^{2}} norm is
Lasso_(statistics)
Mathematical set with an ordering
Associative algebra used in combinatorics Nested set collection Order polytope Ordered field – Algebraic object with an ordered structure Ordered group –
Partially_ordered_set
Mathematical model of the physical space
polytopes, which are the higher-dimensional analogues of polygons and polyhedra. He developed their theory and discovered all the regular polytopes,
Euclidean_geometry
Mathematical invariance under transformations
resonance imaging (fMRI) to compare responses for patterns with symmetrical or random dots. A strong activity was present in extrastriate regions of the occipital
Symmetry
Unsolved problem in computational complexity theory
spaces that contain the two polytopes (not necessarily of the same dimension) which induces a bijection between the polytopes. Manuel Blum and Sampath Kannan (1995)
Graph_isomorphism_problem
Mathematical model combining space and time
Carroll, Sean (2022). The Biggest Ideas in the Universe. New York: Penguin Random House LLC. pp. 155–156. ISBN 978-0-593-18658-9. Curiel, Erik; Bokulich,
Spacetime
Context dependence in quantum measurements
\mathbf {p} } representing a system and the surface of the noncontextuality polytope P {\displaystyle \mathbb {P} } representing all possible noncontextual
Quantum_contextuality
Subfield of convex optimization
maximize or minimize a linear objective function of real variables over a polytope. In semidefinite programming, we instead use real-valued vectors and are
Semidefinite_programming
German mathematician (1826–1866)
Riemann's 1859 paper introducing the complex zeta function "Riemann". Random House Webster's Unabridged Dictionary. Dudenredaktion; Kleiner, Stefan;
Bernhard_Riemann
Graph with tight clique-coloring relation
{\displaystyle x\geq 0} , A x ≤ 1 {\displaystyle Ax\leq 1} form an integral polytope. It is the convex hull of the indicator vectors of independent sets in
Perfect_graph
Decomposition of a graph into hamiltonion cycles
Rosenfeld, Moshe (1986), "On Hamilton decompositions of prisms over simple 3-polytopes", Graphs and Combinatorics, 2 (1): 1–8, doi:10.1007/BF01788070, MR 1117125
Hamiltonian_decomposition
Study of motions and interactions of neutrons
Jaguar – A parallel 3-D Slice Balance Approach transport code for arbitrary polytope grids developed at NNL DANTSYS RAMA – A proprietary 3D method of characteristics
Neutron_transport
Branch of geometry
and Discrete Geometry includes three major branches: general convexity polytopes and polyhedra discrete geometry (though only portions of the latter two
Convex_geometry
Conference in theoretical computer science
Raghavan (2013), Plenary talk 2014 Thomas Rothvoss (2014), "The matching polytope has exponential extension complexity" Shafi Goldwasser (2014), "The Cryptographic
Symposium on Theory of Computing
Symposium_on_Theory_of_Computing
Problems which attempt to find the most efficient way to pack objects into containers
JSTOR 2688954. Betke, Ulrich; Henk, Martin (2000). "Densest lattice packings of 3-polytopes". Computational Geometry. 16 (3): 157–186. arXiv:math/9909172. doi:10
Packing_problems
Israeli mathematician and computer scientist
and for his work on the Hirsch conjecture on the diameter of convex polytopes and in polyhedral combinatorics more generally. Kalai is a noted skeptic
Gil_Kalai
English mathematician (1937–2020)
1 December 1995 Conway, J. H. (1967). "Four-dimensional Archimedean polytopes". Proc. Colloquium on Convexity, Copenhagen. Kobenhavns Univ. Mat. Institut:
John_Horton_Conway
Measure of complexity of real-valued functions
Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq
Rademacher_complexity
Algorithm analysis method
Nina; Ziegler, Günter (1999), "Deformed products and maximal shadows of polytopes", Contemporary Mathematics, vol. 223, American Mathematical Society, pp
Smoothed_analysis
Branch of computer science
computational geometry was the formulation of an algorithm that takes O(n log n). Randomized algorithms that take O(n) expected time, as well as a deterministic algorithm
Computational_geometry
RANDOM POLYTOPE
RANDOM POLYTOPE
Surname or Lastname
English
English : probably a variant of Crandon, a habitational name from Crandon in Somerset or Crandean in Falmer, Sussex. Compare Grandin.
Female
English
Pet form of English Miranda, RANDY means "worthy of admiration."Â Compare with masculine Randy.Â
Female
English
Short form of English Miranda, RANDA means "worthy of admiration."Â
Surname or Lastname
English or Scottish
English or Scottish : unexplained. Possibly, as Black suggests, a reduced form of Langdon.French : from the old Germanic personal name element Lando (see Land), via the oblique case, Landonis.
Boy/Male
English American
Son of Rand.
Male
English
Pet form of English Randall and Randolph, both RANDY means "shield-wolf." Compare with feminine Randy.
Boy/Male
English
Son of Rand.
Female
English
Variant spelling of English Randy, RANDI means "worthy of admiration."
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Male
English
Medieval form of English Randolf, RANDAL means "shield-wolf."
Surname or Lastname
English
English : variant of Ransom.
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Surname or Lastname
English
English : patronymic from Rand 1.
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Male
Scandinavian
 Scandinavian form of Old Norse Randolfr, RANDOLF means "shield-wolf." Compare with another form of Randolf.
Surname or Lastname
English
English : variant spelling of Randall.Americanized spelling of Randel.
Male
English
 Variant spelling of Middle English Randulf, RANDOLF means "shield-wolf." Compare with other forms of Randolf.
Surname or Lastname
English
English : variant of Brandon.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : patronymic from the Middle English personal name Rand(e) (see Rand 1).
Surname or Lastname
English
English : unexplained; perhaps a variant of Francom.
RANDOM POLYTOPE
RANDOM POLYTOPE
Boy/Male
Hindu
Particular
Boy/Male
Indian, Punjabi, Sikh
Divine Lamp
Boy/Male
Hindu, Indian, Marathi, Modern, Punjabi, Sikh
Meaning; Like an Eagle
Girl/Female
Latin American
Victory; triumphant. Famous Bearer: Queen Victoria.
Boy/Male
Tamil
Sudarsan | ஸà¯à®¤à®°à¯à®¸à®¨Â
Lord Perumal, Good looking, Lion, Vishnus weapon
Boy/Male
English
From the flooding brook.
Boy/Male
Indian
Honor of the state
Male
Hungarian
Hungarian form of Latin Gustavus, GUSZTÃV means "meditation staff."
Female
Hindi/Indian
Hindi name RADHIKA means "fulfiller of desires." In mythology, this is an epithet belonging to Radha.Â
Boy/Male
Tamil
RANDOM POLYTOPE
RANDOM POLYTOPE
RANDOM POLYTOPE
RANDOM POLYTOPE
RANDOM POLYTOPE
n.
Ransom.
v. i.
To wander at random; to scatter.
n.
Distance to which a missile is cast; range; reach; as, the random of a rifle ball.
n.
Extra hazard; chance; accident; random.
n.
A roving motion; course without definite direction; want of direction, rule, or method; hazard; chance; -- commonly used in the phrase at random, that is, without a settled point of direction; at hazard.
n.
To redeem from captivity, servitude, punishment, or forfeit, by paying a price; to buy out of servitude or penalty; to rescue; to deliver; as, to ransom prisoners from an enemy.
adv.
At random; hit or miss. (Obs.)
n.
To exact a ransom for, or a payment on.
n.
Random.
p. pr. & vb. n.
of Ransom
adv.
In a random manner.
v. i.
To extend or grow at random.
a.
Cruising at random on the ocean.
v. i.
To go or stray at random.
n.
Ransom; release.
n.
Anything driven at random.
imp. & p. p.
of Ransom
a.
Going at random or by chance; done or made at hazard, or without settled direction, aim, or purpose; hazarded without previous calculation; left to chance; haphazard; as, a random guess.
n.
The release of a captive, or of captured property, by payment of a consideration; redemption; as, prisoners hopeless of ransom.