Blaschke selection theorem

The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence

{

K

n

}

{\displaystyle \{K_{n}\}}

of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence

{

K

n

m

}

{\displaystyle \{K_{n_{m}}\}}

and a convex set

K

{\displaystyle K}

such that

K

n

m

{\displaystyle K_{n_{m}}}

converges to

K

{\displaystyle K}

in the Hausdorff metric. The theorem is named for Wilhelm Blaschke.

Alternate statements

A succinct statement of the theorem is that the metric space of convex bodies is locally compact.

Using the Hausdorff metric on sets, every infinite collection of compact subsets of the unit ball has a limit point (and that limit point is itself a compact set).

Application

As an example of its use, the isoperimetric problem can be shown to have a solution. That is, there exists a curve of fixed length that encloses the maximum area possible. Other problems likewise can be shown to have a solution:

Lebesgue's universal covering problem for a convex universal cover of minimal size for the collection of all sets in the plane of unit diameter,

the maximum inclusion problem,

and the Moser's worm problem for a convex universal cover of minimal size for the collection of planar curves of unit length.

Notes

References

A. B. Ivanov (2001) [1994], "Blaschke selection theorem", Encyclopedia of Mathematics, EMS Press

V. A. Zalgaller (2001) [1994], "Metric space of convex sets", Encyclopedia of Mathematics, EMS Press

Kai-Seng Chou; Xi-Ping Zhu (2001). The Curve Shortening Problem.