Math 2.0

Convex and discrete geometry Convex sets: basic notions Theorem

Blaschke selection theorem

Builds on Convex setCompact setBounded setSubsequenceHausdorff distance

Statement

Let n≥1n \ge 1 and let (Km)m≥1(K_m)_{m \ge 1} be a sequence of nonempty compact convex sets in ℝn\R^n, all contained in one bounded set. Then (Km)(K_m) has a subsequence that converges, in the Hausdorff distance, to a nonempty compact convex set K⊆ℝnK \subseteq \R^n.

Remarks

The nonempty compact subsets of a compact metric space form a compact metric space under the Hausdorff distance. All KmK_m lie in a closed ball, which is compact, so a subsequence converges to some nonempty compact set KK. A Hausdorff limit of convex sets is convex, which gives the theorem. The limit can be smaller in dimension than the KmK_m: the rectangles Km=[0,1]×[0,1m]K_m = [0, 1] \times [0, \frac1m] converge to the segment [0,1]×{0}[0,1] \times \set{0}. Boundedness is needed: the segments [0,m]×{0}[0, m] \times \set{0} have no convergent subsequence, since their Hausdorff distances to any fixed compact set tend to infinity.

Going deeper

For a student

The theorem makes the space of convex bodies with the Hausdorff metric locally compact. It is used to prove the existence of extremal bodies. A continuous functional, such as volume or mixed volume, attains its maximum and minimum over a bounded closed family of convex sets. Existence proofs for the John ellipsoid and for solutions of isoperimetric-type problems use it in this way.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Bounded set, concept, no evidence levelBounded setconceptCompact set, concept, no evidence levelCompact setconceptConvex set, concept, no evidence levelConvex setconceptHausdorff distance, concept, no evidence levelHausdorff distanceconceptSubsequence, concept, no evidence levelSubsequenceconceptBlaschke selection theorem, statement, no evidence levelBlaschke selection th…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. Bounded set concept
  2. Compact set concept
  3. Convex set concept
  4. Hausdorff distance concept
  5. Subsequence concept
  6. Blaschke selection theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Blaschke_selection_theorem
About this page
TagD9C581V5HG
URIm2://D9C581V5HG
Kindstatement, theorem
WritingW0 stub
MSC52A20, 54B20
StatusThis copy of the site has no build data. Run m2 status D9C581V5HG.
AboutConvex set, Hausdorff distance
UsesConvex set, Compact set, Bounded set, Subsequence, Hausdorff distance
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