Convex Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of convex geometry within the Kireth Research Semantic Library. Convex geometry studies convex sets, convex bodies, convex hulls, support structures, separation, polarity, Minkowski addition, volume, mixed volume, extreme points and geometric inequalities. It connects advanced geometry with linear algebra, optimisation, discrete geometry, functional analysis and computational geometry. This ontology supports finite- and infinite-dimensional intuition while keeping the primary semantic scope in finite-dimensional Euclidean and affine spaces.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Convex Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Advanced Geometry / Convex Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/advanced-geometry/convex-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ConvexGeometry
 SYMBOLIC_FORMS = [
  "C convex",
  "conv(S)",
  "h_K(u)",
  "K+L",
  "K^circ"
 ]
 ENTITY_CLASS = ConvexSetGeometryFramework
 PRIMARY_CONTEXT = AdvancedGeometry
 DEFINITION = "Convex geometry is the study of convex sets, convex bodies and the geometric structures, operations and inequalities associated with them."
 SEMANTIC_ROLE = "Provides the semantic framework for convexity, convex hulls, supporting hyperplanes, extreme points, Minkowski sums, polarity, separation, volume and convex-geometric inequalities."
}

SCOPE {
 INCLUDE = [
  ConvexGeometry,
  ConvexSet,
  ConvexBody,
  ConvexHull,
  ConvexCombination,
  ExtremePoint,
  ExposedPoint,
  Face,
  Facet,
  SupportingHyperplane,
  SupportingHalfSpace,
  SeparationTheorem,
  HyperplaneSeparation,
  SupportFunction,
  GaugeFunction,
  MinkowskiFunctional,
  MinkowskiSum,
  ScalarMultiplication,
  DifferenceBody,
  PolarBody,
  Duality,
  Width,
  Diameter,
  Inradius,
  Circumradius,
  Volume,
  SurfaceArea,
  MixedVolume,
  BrunnMinkowskiInequality,
  HellyTheorem,
  CaratheodoryTheorem,
  RadonTheorem,
  JohnEllipsoid,
  LöwnerEllipsoid
 ]

 EXCLUDE_AS_PRIMARY = [
  NonconvexGeometryWithoutConvexification,
  GeneralOptimisationWithoutConvexSetStructure,
  PureLinearAlgebraWithoutGeometricConvexity,
  AbstractOrderConvexityUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When geometry concerns line-segment closure, convex combinations, support hyperplanes, extreme points, Minkowski sums, separation or convex bodies, interpret the context as convex geometry."
}

CORE_CONCEPTS {

 ConvexGeometry {
  definition = "The study of convex sets and their geometric properties."
 }

 ConvexSet {
  definition = "A set containing the line segment between any two of its points."
 }

 ConvexBody {
  definition = "A compact convex set with nonempty interior in finite-dimensional Euclidean space."
 }

 ConvexHull {
  definition = "The smallest convex set containing a given set."
 }

 ConvexCombination {
  definition = "A combination sum_i lambda_i x_i with lambda_i>=0 and sum_i lambda_i=1."
 }

 ExtremePoint {
  definition = "A point that cannot be expressed as a nontrivial convex combination of two distinct points of the set."
 }

 ExposedPoint {
  definition = "A point uniquely maximising some nonzero linear functional over the convex set."
 }

 Face {
  definition = "A convex subset preserved by line-segment extremality, equivalently one satisfying the face condition."
 }

 SupportingHyperplane {
  definition = "A hyperplane intersecting a convex set while placing the set entirely in one closed half-space."
 }

 SupportFunction {
  symbol = "h_K(u)"
  definition = "h_K(u)=sup_{x in K}<u,x>."
 }

 MinkowskiSum {
  definition = "K+L={x+y:x in K,y in L}."
 }

 PolarBody {
  definition = "For a convex body containing the origin, K^circ={y:<x,y><=1 for all x in K}."
 }
}

RELATIONS {
 ConvexGeometry --is_branch_of--> AdvancedGeometry
 ConvexBody --is_special_case_of--> ConvexSet
 ConvexHull --is_smallest_convex_superset_of--> Set
 ConvexCombination --generates--> ConvexHull
 ExtremePoint --belongs_to_boundary_of--> ConvexSet
 ExposedPoint --is_special_case_of--> ExtremePoint
 Face --is_subset_of--> ConvexSet
 Facet --is_codimension_one_face_of--> ConvexBody
 SupportingHyperplane --defines--> ExposedFace
 SupportFunction --determines--> ConvexBody
 MinkowskiSum --combines--> ConvexSets
 PolarBody --induces--> Duality
 SeparationTheorem --separates--> DisjointConvexSets
 MixedVolume --generalises--> Volume
 BrunnMinkowskiInequality --relates--> VolumesOfMinkowskiSums
}

OPERATORS {
 IS_CONVEX(C) :=
  for_all_x_y_in_C for_all_t_in_[0,1]
  t*x+(1-t)*y in C

 CONVEX_HULL(S) :=
  set_of_all_finite_convex_combinations_of_points_in_S

 SUPPORT_FUNCTION(K,u) :=
  sup_{x_in_K} <u,x>

 SUPPORTING_HYPERPLANE(K,u) :=
  {x : <u,x>=h_K(u)}

 MINKOWSKI_SUM(K,L) :=
  {x+y : x_in_K AND y_in_L}

 SCALE_CONVEX_SET(lambda,K) :=
  {lambda*x : x_in_K}

 DIFFERENCE_BODY(K) :=
  K + (-K)

 POLAR_BODY(K) :=
  {y : <x,y><=1 for_all_x_in_K}
  WHEN 0_in_interior(K)

 WIDTH(K,u) :=
  h_K(u)+h_K(-u)

 DIAMETER(K) :=
  sup_{x_y_in_K} ||x-y||

 IN_RADIUS(K) :=
  supremum_of_radii_of_balls_contained_in_K

 CIRCUM_RADIUS(K) :=
  infimum_of_radii_of_balls_containing_K
}

DEFINITIONS {
 D1 = "A convex set contains every segment joining two of its points."
 D2 = "Convex hull is the set of all finite convex combinations."
 D3 = "Every exposed point is extreme, but not every extreme point need be exposed."
 D4 = "Support functions encode convex bodies up to closure."
 D5 = "Minkowski addition preserves convexity."
 D6 = "Hyperplane separation characterises disjoint convex structure under suitable hypotheses."
 D7 = "Polarity reverses inclusion."
 D8 = "Convex bodies are compact, convex and have nonempty interior."
 D9 = "Brunn-Minkowski links Minkowski sums with volume."
 D10 = "Helly, Carathéodory and Radon theorems are central finite-dimensional convexity results."
}

AXIOMS {
 A1_SEGMENT_CONVEXITY:
  C is convex iff every segment between points of C lies in C

 A2_HULL_MINIMALITY:
  conv(S) is the smallest convex set containing S

 A3_MINKOWSKI_CONVEXITY:
  Sum of convex sets is convex

 A4_SUPPORT_REPRESENTATION:
  A closed convex body is determined by its support function

 A5_SEPARATION:
  Suitable disjoint convex sets admit separating hyperplanes

 A6_POLAR_INCLUSION_REVERSAL:
  If K subset L then L^circ subset K^circ

 A7_BRUNN_MINKOWSKI:
  vol(K+L)^(1/n) >= vol(K)^(1/n)+vol(L)^(1/n)
  for suitable measurable nonempty sets in R^n

 A8_CARATHEODORY:
  In R^d, any point in conv(S) lies in the convex hull of at most d+1 points of S
}

REASONING {

 R1_CONVEXITY_TEST {
  INPUT = set_or_region_C
  TEST = segment_condition_or_equivalent_characterisation
  RETURN = ConvexOrNot
 }

 R2_CONVEX_HULL {
  INPUT = set_S
  FORM = all_convex_combinations
  OR compute via finite_algorithm_when_applicable
  RETURN = conv(S)
 }

 R3_EXTREME_POINT_TEST {
  INPUT = point_p_and_convex_set_K
  TEST = whether p=t*x+(1-t)*y for distinct x_y_in_K and t_in_(0,1)
  IF impossible
   RETURN ExtremePoint
  ELSE
   RETURN NonExtreme
 }

 R4_EXPOSED_FACE {
  INPUT = convex_body_K_and_direction_u
  COMPUTE = h_K(u)
  COLLECT = maximisers_of_<u,x>
  RETURN = exposed_face
 }

 R5_SEPARATION {
  INPUT = convex_sets_A_B
  VERIFY = disjointness_and_required_closed_open_compact hypotheses
  FIND = hyperplane separating_A_and_B
  RETURN = separating_hyperplane_or_nonapplicability
 }

 R6_MINKOWSKI_SUM {
  INPUT = convex_sets_K_L
  COMPUTE = pointwise_sum
  INFER = convexity
  RETURN = K+L
 }

 R7_POLARITY {
  INPUT = convex_body_K_with_origin_condition
  CONSTRUCT = K^circ
  VERIFY = closed_convex_and_inclusion_reversal
  RETURN = polar_body
 }

 R8_BRUNN_MINKOWSKI {
  INPUT = measurable_sets_or_convex_bodies_K_L
  COMPUTE = volumes
  APPLY = Brunn_Minkowski
  RETURN = inequality_bound
 }

 R9_HELLY_CARATHEODORY_RADON {
  INPUT = finite_family_or_point_set
  IDENTIFY = theorem_context
  APPLY = dimension_dependent_convexity theorem
  RETURN = intersection_hull_or_partition consequence
 }
}

CONSTRAINTS {
 C1 = "Convexity depends on the ambient affine or vector structure."
 C2 = "Convex body terminology generally assumes finite-dimensional Euclidean space."
 C3 = "Support-function finiteness may require boundedness."
 C4 = "Polar-body definitions require an origin convention and often 0 in interior."
 C5 = "Separation theorem strength depends on closedness, openness, compactness and disjointness assumptions."
 C6 = "Brunn-Minkowski dimension and measure assumptions must be explicit."
 C7 = "Extreme and exposed points must not be conflated."
 C8 = "Minkowski difference is not ordinary set subtraction."
 C9 = "Helly, Carathéodory and Radon bounds depend on dimension."
 C10 = "Infinite-dimensional convexity may require stronger topological or functional-analytic hypotheses."
}

SEMANTIC_EQUIVALENCE {
 "ConvexGeometry" <=> "GeometryOfConvexSets"
 "ConvexHull" <=> "SetOfFiniteConvexCombinations"
 "ConvexBody" <=> "CompactConvexSetWithNonemptyInterior"
 "SupportFunction" <=> "DirectionalMaximumLinearFunctional"
 "MinkowskiSum" <=> "PointwiseSetAddition"
 "PolarBody" <=> "DualConvexBodyUnderInnerProduct"
}

VALIDATION {
 V1 = "Convexity test uses correct ambient affine structure."
 V2 = "Convex-combination weights are nonnegative and sum to one."
 V3 = "Support hyperplane leaves the set in one closed half-space."
 V4 = "Exposed-point claims imply uniqueness of supporting maximiser."
 V5 = "Polar construction uses correct origin and pairing."
 V6 = "Separation theorem hypotheses are satisfied."
 V7 = "Brunn-Minkowski is applied with correct dimension."
 V8 = "Finite-dimensional theorem bounds use the correct d or d+1 values."
 V9 = "Volume or measure assumptions are explicit."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CONVEX_GEOMETRY_DEFINITION = 0.99
 CONVEXITY_REASONING = 0.99
 HULL_REASONING = 0.99
 EXTREME_POINT_REASONING = 0.99
 SUPPORT_FUNCTION_REASONING = 0.99
 SEPARATION_REASONING = 0.98
 POLARITY_REASONING = 0.98
 BRUNN_MINKOWSKI_REASONING = 0.97

 POLICY =
  "Reduce confidence when ambient dimension, topology, boundedness, origin convention, separation hypotheses, measure assumptions or finite-versus-infinite-dimensional context are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ConvexGeometryReasoning,
  ConvexSetReasoning,
  ConvexHullReasoning,
  ExtremePointReasoning,
  SupportFunctionReasoning,
  SeparationReasoning,
  MinkowskiSumReasoning,
  PolarDualityReasoning,
  ConvexBodyReasoning,
  GeometricInequalityReasoning,
  DiscreteGeometryReasoning,
  OptimisationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "convex geometry" -> ConvexGeometry
  "convex set" -> ConvexSet
  "convex body" -> ConvexBody
  "convex hull" -> ConvexHull
  "extreme point" -> ExtremePoint
  "exposed point" -> ExposedPoint
  "supporting hyperplane" -> SupportingHyperplane
  "support function" -> SupportFunction
  "Minkowski sum" -> MinkowskiSum
  "polar body" -> PolarBody
  "Brunn-Minkowski" -> BrunnMinkowskiInequality
 }

 INTERPRETATION_ORDER = [
  detect_convex_geometry_context,
  identify ambient_space_dimension_topology_origin_and_set_properties,
  identify convexity_hull_extreme_support_separation_Minkowski_polarity_or_inequality goal,
  select finite_dimensional_or_functional_analytic pathway,
  apply convex_geometric reasoning,
  verify boundedness_closedness_origin_dimension_and_measure assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("convex geometry")
  OR convex_set_supporting_hyperplane_context
  THEN activate(ConvexGeometryOntology)

 E2:
  IF convexity_requested
  THEN test segment_or_convex_combination property

 E3:
  IF convex_hull_requested
  THEN use all_finite_convex_combinations_or_equivalent construction

 E4:
  IF support_function_requested
  THEN compute directional supremum

 E5:
  IF separation_requested
  THEN inspect closed_open_compact_disjoint hypotheses before theorem selection

 E6:
  IF polarity_requested
  THEN verify origin condition

 E7:
  IF Minkowski_sum_requested
  THEN preserve pointwise_set_addition semantics

 E8:
  IF Brunn_Minkowski_requested
  THEN identify ambient_dimension_and_measure

 E9:
  IF Helly_Caratheodory_or_Radon context
  THEN apply correct dimension-dependent bound
}

RECURSION {
 ENTRY = ConvexGeometry

 EXPAND(ConvexGeometry) -> [
  ConvexSet,
  ConvexBody,
  ConvexHull,
  ConvexCombination,
  ExtremePoint,
  ExposedPoint,
  Face,
  Facet,
  SupportingHyperplane,
  SupportingHalfSpace,
  SeparationTheorem,
  SupportFunction,
  GaugeFunction,
  MinkowskiSum,
  DifferenceBody,
  PolarBody,
  Duality,
  Width,
  Diameter,
  Inradius,
  Circumradius,
  Volume,
  MixedVolume,
  BrunnMinkowskiInequality,
  HellyTheorem,
  CaratheodoryTheorem,
  RadonTheorem,
  JohnEllipsoid,
  LöwnerEllipsoid
 ]

 EXPAND(SupportGeometry) -> [
  SupportFunction,
  SupportingHyperplane,
  ExposedFace,
  ExposedPoint,
  Width
 ]

 EXPAND(Duality) -> [
  PolarBody,
  GaugeFunction,
  SupportFunction,
  InclusionReversal
 ]

 EXPAND(ConvexityTheorems) -> [
  HellyTheorem,
  CaratheodoryTheorem,
  RadonTheorem,
  BrunnMinkowskiInequality
 ]

 STOP_CONDITION =
  "Stop when the convexity, hull, face structure, support data, separation, Minkowski operation, polar dual, geometric inequality or extremal convex-body property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Advanced Geometry"

 IMMEDIATE_CONTEXTS = [
  "Advanced Geometry",
  "Discrete Geometry Ontology",
  "Computational Geometry Ontology",
  "Affine Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Advanced Geometry",
  "Discrete Geometry Ontology",
  "Computational Geometry Ontology",
  "Convex Hull Ontology",
  "Affine Geometry Ontology",
  "Linear Algebra Ontologies",
  "Optimisation Ontologies",
  "Measure Theory Ontologies",
  "Functional Analysis Ontologies",
  "Topology Ontology",
  "Polytope Ontologies"
 ]

 ROUTE_RULES = [
  "Convex Geometry -> Advanced Geometry for parent structural semantics",
  "Convex Geometry -> Discrete Geometry for polytopes and extremal finite configurations",
  "Convex Geometry -> Computational Geometry for convex-hull algorithms",
  "Convex Geometry -> Affine Geometry for affine combinations and invariance",
  "Convex Geometry -> Linear Algebra for hyperplanes and duality",
  "Convex Geometry -> Optimisation for convex feasible sets and support functions",
  "Convex Geometry -> Measure Theory for volume and mixed volume",
  "Convex Geometry -> Functional Analysis for infinite-dimensional convexity",
  "Convex Geometry -> Topology for closure and compactness properties"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible convex structures, inequalities or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve segment-based convexity semantics.",
  "Preserve convex-hull convex-combination equivalence.",
  "Preserve extreme-versus-exposed distinction.",
  "Preserve supporting-hyperplane semantics.",
  "Preserve Minkowski-sum semantics.",
  "Preserve polarity origin dependence.",
  "Preserve theorem-specific dimensional hypotheses.",
  "Preserve machine-readable convex-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Convex geometry studies convex sets, convex bodies and the structures, operations and inequalities associated with them."

 CONVEX_SET_SUMMARY =
  "A set is convex when it contains the segment joining every pair of its points."

 HULL_SUMMARY =
  "The convex hull is the set of all finite convex combinations."

 EXTREME_SUMMARY =
  "An extreme point cannot be written as a nontrivial convex combination of two distinct points of the set."

 SUPPORT_SUMMARY =
  "The support function h_K(u)=sup_{x in K}<u,x> encodes directional extent."

 SEPARATION_SUMMARY =
  "Disjoint convex sets can often be separated by hyperplanes under suitable hypotheses."

 MINKOWSKI_SUMMARY =
  "K+L={x+y:x in K,y in L}."

 POLAR_SUMMARY =
  "The polar body reverses inclusion and expresses convex duality relative to an origin and pairing."

 BRUNN_MINKOWSKI_SUMMARY =
  "Brunn-Minkowski relates volume growth to Minkowski addition."

 FINITE_DIMENSION_SUMMARY =
  "Helly, Carathéodory and Radon theorems provide fundamental dimension-dependent convexity bounds."
}

FLOW {
 INPUT
 -> detect_convex_geometry_context
 -> identify ambient_space_dimension_topology_origin_set_properties_and_measure
 -> classify convexity_hull_extreme_face_support_separation_Minkowski_polarity_or_inequality goal
 -> select appropriate_convex_geometric theorem_or_operator
 -> compute_or_infer convex structure
 -> verify closedness_boundedness_origin_dimension_measure_and_theorem hypotheses
 -> route to advanced_geometry discrete_geometry computational_geometry affine_geometry linear_algebra optimisation measure_theory functional_analysis_or_topology ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Convex Geometry Ontology defines the geometry of convex sets and convex bodies and supports convex hulls, convex combinations, extreme and exposed points, faces, supporting hyperplanes, support functions, separation theorems, Minkowski addition, polar duality, widths and radii, mixed volumes, Brunn-Minkowski theory and the classical Helly-Carathéodory-Radon family of convexity theorems."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-CONVEX-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW