---
tag: D9C581V5HG
kind: statement
role: theorem
name: Blaschke selection theorem
msc: [52A20, 54B20]
uses: [02AS45T3H9, 0R6EWFKRCF, 09QSZVVM26, 0F1PR74BRX, F79WYYC07C]
about: [02AS45T3H9, F79WYYC07C]
sources:
  - url: https://en.wikipedia.org/wiki/Blaschke_selection_theorem
same_as:
  wikidata: Q784049
  wikipedia: Blaschke selection theorem
---

## Statement

Let $n \ge 1$ and let $(K_m)_{m \ge 1}$ be a sequence of nonempty [[compact|0R6EWFKRCF]] [[convex sets|02AS45T3H9]] in $\R^n$, all contained in one [[bounded set|09QSZVVM26]].
Then $(K_m)$ has a [[subsequence|0F1PR74BRX]] that converges, in the [[Hausdorff distance|F79WYYC07C]], to a nonempty compact convex set $K \subseteq \R^n$.

## Remarks

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

## Explanations

### L1 human

Take infinitely many convex shapes that all fit in one box.
Then one can pick some of them, one after another, that settle down to a limiting convex shape.
This is the Bolzano–Weierstrass theorem for convex shapes instead of numbers.

### L2 human

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|QB3Q5FBJ08]], attains its maximum and minimum over a bounded closed family of convex sets.
Existence proofs for the [[John ellipsoid|WNHXE7FCG4]] and for solutions of isoperimetric-type problems use it in this way.