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Research draft

convex set

vr.tr.convex-set · XCT.QLT

Let an agent explain convex sets and their properties, describe key examples and theorems, support recognising and using convexity in optimisation, and support learning.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain convex sets and their properties, describe key examples and theorems, support recognising and using convexity in optimisation, and support learning.

A set in a real vector space such that the line segment between any two of its points lies entirely within the set, including half-spaces, balls and hyperballs, convex polytopes, parallelotopes, spectrahedra and bodies of constant brightness; convex sets are fundamental in geometry, functional analysis and optimisation, where convexity guarantees that local optima are global, and are characterised by supporting hyperplanes, extreme points and separation theorems.

What it is for: Sets containing all segments between their points.

It can be explain properties; describe examples and theorems; support use in optimisation; support learning.

Distinguishing features

Segment property

Supporting hyperplanes

Extreme points

Optimisation guarantees

What it looks like

Not physical; shapes without indentations.

How it is recognised

Contains every segment between its points

Balls, half-spaces, polytopes

Star domains are convex from one point only; non-convex sets have indentations

Related models

is a kind of - category

star domain

is a kind of - category

topological subspace

is related to - convex sets in the plane

plane

is related to - convex inequalities

inequality

In practice

Families and kinds

half-spaces and hyperplanes

balls and ellipsoids

convex polytopes and parallelotopes

spectrahedra and cones

convex bodies such as bodies of constant width

Standards and regulation

Mathematical conventions

No regulation

Failure modes and hazards

Confusing convex sets with convex functions

Assuming convexity without checking

Misapplying theorems in infinite dimensions

Also called

hyperballspectrahedronconvex polytopehalf-spaceparallelotopebody of constant brightnessconvex hulllinear matrix inequalityconvex coneopen ballabsolutely convex setconvex bodygeodesic convexityEquiprojective polyhedraHanner polytopeneighborly polytopezonotopehypersimplexstable set polytopeassociahedronorder polytopeMatching polytopeconical hullopen unit ballBarrelled setomnitruncated 5-cell

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Understand What convex sets are.

Mathematics.

Definition

Definition and properties.

Definition

Definition.

  1. What is a convex set, and what properties such as closure under intersection and convex hulls follow? definition
  2. Is a convex set or a convex function meant? boundary

Examples

Examples.

Examples

Examples.

  1. What are half-spaces, balls, polytopes, spectrahedra and other key convex sets? definition
  2. Which entry fits a specific example? action
Theory Theorems.

Analysis.

Theorems

Key theorems.

Theorems

Theorems.

  1. What do separation, supporting hyperplane, Krein-Milman and Caratheodory theorems say? definition
  2. Which entry fits functional analysis? action

Structure

Extreme points and faces.

Structure

Structure.

  1. How are extreme points, faces and dimension of convex sets analysed? definition
  2. Which entry fits polytopes? action
Apply Applications.

Practice.

Optimisation

Convex optimisation.

Optimisation

Optimisation.

  1. Why does convexity make optimisation tractable, and how is this used? provenance
  2. Which entry fits convex optimisation? action

Check

Checking convexity.

Check

Checking.

  1. Is this set convex, and how can convexity be verified? action
  2. Which entry fits geometry software? action
Learn History and teaching.

Education.

History

History.

History

History.

  1. How did convex geometry develop from Minkowski onward? provenance
  2. Which references are standard? provenance

Teach

Teaching.

Teach

Teaching.

  1. How can convexity be taught with pictures and examples? action
  2. Which misconceptions arise? provenance

What the second pass must settle

  • Should convex functions be a separate entry?
  • How should optimisation resources be linked?
  • How should geometry textbooks be linked?
Draft as JSON