The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence
{
K
n
}
of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence
{
K
n
m
}
and a convex set
K
such that
K
n
m
converges to
K
in the Hausdorff metric. The theorem is named for Wilhelm Blaschke.
A succinct statement of the theorem is that a 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).
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.
