Math 2.0

Convex and discrete geometry Convex geometric analysis and geometric inequalities Theorem

Blaschke–Lebesgue theorem

Builds on Curve of constant widthReuleaux triangleBarbier's theorem

Statement

Let w>0w > 0, and let KK be a plane convex body of constant width ww. So KK is bounded by a curve of constant width ww. Then the area of KK is at least π−32w2\frac{\pi - \sqrt{3}}{2} w^2. Equality holds if and only if KK is a Reuleaux triangle of width ww.

Remarks

One classical route approximates KK by Reuleaux polygons of width ww, which are bodies of constant width bounded by finitely many circular arcs of radius ww. For Reuleaux polygons one shows that the area can be lowered by deforming the polygon towards a Reuleaux triangle, and a limit argument finishes the proof. Example: a disc of diameter ww has area π4w2≈0.785 w2\frac{\pi}{4} w^2 \approx 0.785 \, w^2, while the Reuleaux triangle has area π−32w2≈0.705 w2\frac{\pi - \sqrt{3}}{2} w^2 \approx 0.705 \, w^2. The constant width hypothesis is needed: a segment of length ww has diameter ww and area 00.

Going deeper

For a student

The maximum among bodies of constant width ww is the disc, which follows from Barbier's theorem and the isoperimetric inequality, since all such bodies have perimeter πw\pi w. The minimum is much harder, and the Blaschke–Lebesgue theorem identifies it. In three dimensions the question of the body of constant width with least volume is open.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

Show the path as a graph
Arc length, concept, no evidence levelArc lengthconceptCircle, concept, no evidence levelCircleconceptConstruction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptCurve of constant width, concept, no evidence levelCurve of constant wid…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectIntegral geometry, concept, no evidence levelIntegral geometryconceptLine, concept, no evidence levelLineconceptOrdered field, definition, no evidence levelOrdered fielddefinitionPlane curve, concept, no evidence levelPlane curveconceptProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectReuleaux triangle, example, no evidence levelReuleaux triangleexampleSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionGeometric probability, concept, no evidence levelGeometric probabilityconceptBuffon's needle problem, problem, no evidence levelBuffon's needle probl…problemCrofton formula, statement, no evidence levelCrofton formulastatementBarbier's theorem, statement, no evidence levelBarbier's theoremstatementBlaschke–Lebesgue theorem, statement, no evidence levelBlaschke–Lebesgue the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Arc length concept
  2. Circle concept
  3. Construction of the reals from Cauchy sequences concept
  4. Curve of constant width concept
  5. Element definition
  6. Integers object
  7. Integral geometry concept
  8. Line concept
  9. Ordered field definition
  10. Plane curve concept
  11. Proposition concept
  12. Random experiment concept
  13. Rational numbers object
  14. Reuleaux triangle example
  15. Set definition
  16. Ring of integers statement
  17. Disjunction definition
  18. Implication definition
  19. Predicate definition
  20. Union definition
  21. Universal quantifier definition
  22. Sample space definition
  23. The rationals form a field statement
  24. Subset definition
  25. The rationals form an ordered field statement
  26. Power set definition
  27. Universal set definition
  28. Order on the reals definition
  29. Complement definition
  30. Interval definition
  31. Sigma-algebra definition
  32. Event definition
  33. Probability axioms concept
  34. Probability measure definition
  35. Probability space definition
  36. Geometric probability concept
  37. Buffon's needle problem problem
  38. Crofton formula statement
  39. Barbier's theorem statement
  40. Blaschke–Lebesgue theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Blaschke%E2%80%93Lebesgue_theorem
About this page
TagXVNB6HV2Y1
URIm2://XVNB6HV2Y1
Kindstatement, theorem
WritingW0 stub
MSC52A10, 52A40, 52A38
StatusThis copy of the site has no build data. Run m2 status XVNB6HV2Y1.
AboutCurve of constant width
UsesCurve of constant width, Reuleaux triangle, Barbier's theorem
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