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Convex hull

The smallest convex set containing a given set, equivalently all finite convex combinations of its points.

Version
v2 · 2026-09-19 · History
Domain-specific #
3904
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Convex Geometry → Mathematics

Core Idea

The convex hull conv(S) is the intersection of every convex set containing S and in finite-dimensional spaces consists of weighted finite sums of points in S with nonnegative weights totaling one. Closing under line segments repeatedly fills every admissible mixture while discarding no more points than convexity requires; extreme points and supporting hyperplanes characterize its boundary. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

Rubber Band Shape

Put some pegs in a board and stretch a rubber band around all of them, then let it snap tight. The shape inside the rubber band, filled in, is the convex hull of the pegs. It's the smallest no-dent shape that holds every peg.

Smallest No-Dent Wrapper

A shape is convex if the straight line between any two of its points stays inside it. The convex hull of a group of points is the smallest convex shape that contains all of them. You can build it by drawing lines between points, then lines between points on those lines, and so on, filling everything in. For dots on paper, it's like stretching a rubber band around them. The corners of the hull are always some of the original points.

Minimal Convex Enclosure

The convex hull of a set S is the smallest convex set that contains S, where a set is convex if the line segment between any two of its points lies inside it. One way to define it is as the intersection of all convex sets containing S. Another, in ordinary finite-dimensional space, is as the set of all weighted averages of finitely many points of S, where the weights are nonnegative and add up to one; these are called convex combinations. You can think of it as filling in every line segment between points, and repeating, until nothing more needs to be added. Its corner points, called extreme points, and the flat surfaces that touch it from outside, called supporting hyperplanes, describe its boundary. The hull must both contain S and be no bigger than convexity requires.

 

The convex hull conv(S) of a subset S of a real affine or vector space is the intersection of every convex set containing S, and so is the smallest convex superset of S. In finite-dimensional spaces it coincides with the set of convex combinations of S: finite sums of points of S with nonnegative weights totaling one. Constructively, closing S under line segments, repeatedly if necessary, fills every admissible mixture while discarding no more than convexity requires. The boundary is characterized by extreme points, which cannot be written as nontrivial convex combinations, and by supporting hyperplanes, which touch the hull while leaving it on one side. The identity rests on fixing the ambient space and scalar field and on two conditions together: conv(S) contains S, and it is minimal among convex sets containing S.

Scope of Application

Convex hull belongs to convex geometry and is useful where the analyst can specify the typed convex geometry carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the ambient affine space and scalar field are fixed and the result both contains the source set and is minimal among convex containing sets. The scope is broad within that domain but bounded by the need for the ambient affine space and scalar field are fixed and the result both contains the source set and is minimal among convex containing sets. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the ambient affine space and scalar field are fixed and the result both contains the source set and is minimal among convex containing sets the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Convex hull can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Convex hull. Convex hull compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

  1. Identify the carrier. State what the elements, states, objects, or observations are: the typed convex geometry carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
  2. Lock the constitutive rule. Express the ambient affine space and scalar field are fixed and the result both contains the source set and is minimal among convex containing sets independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of convex geometry because they reuse the typed convex geometry carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Closing under line segments repeatedly fills every admissible mixture while discarding no more points than convexity requires; extreme points and supporting hyperplanes characterize its boundary., and type the carrier, state every parameter and convention in the definition, test that the ambient affine space and scalar field are fixed and the result both contains the source set and is minimal among convex containing sets, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Local relationship map for Convex hullParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Convex hullDOMAINPrime abstraction: Convexity — is a kind ofConvexityPRIMEDomain-specific abstraction: Newton polytope — is a kind ofNewton polytopeDOMAIN

Current abstraction Convex hull Domain-specific

Parents (1) — outgoing typed relations

  • Convex hull is a kind of Convexity Prime

    What makes it its own entry: the domain-specific identity determined by the ambient affine space and scalar field are fixed and the result both contains the source set and is minimal among convex containing sets.

Children (1) — abstractions linked here by typed relations

  • Newton polytope Domain-specific is a kind of Convex hull

    A Newton polytope is the convex hull of a polynomial's nonzero-term exponent vectors.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Convex hull sits in a crowded region of the domain-specific corpus (4th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Convex Analysis & Optimization Structures (8 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25