Blaschke selection theorem

<h2 id="alternate-statements">Alternate statements</h2>
<ul><li>A succinct statement of the theorem is that the <a href="/facts/Metric_space/gnBBRXtc">metric space</a> of convex bodies is <a href="/facts/Locally_compact/hesalT7q">locally compact</a>.</li>
<li>Using the <a href="/facts/Hausdorff_metric/3tKbFmKu">Hausdorff metric</a> on sets, every infinite collection of compact subsets of the unit ball has a limit point (and that limit point is itself a <a href="/facts/Compact_set/d0cgXJH7">compact set</a>).</li></ul>
<h2 id="application">Application</h2>
<p>As an example of its use, the <a href="/facts/Isoperimetric_inequality/u02PG11C">isoperimetric problem</a> can be shown to have a solution.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  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:
</p>
<ul><li><a href="/facts/Lebesgue%27s_universal_covering_problem/p8gnf8Qm">Lebesgue's universal covering problem</a> for a convex universal cover of minimal size for the collection of all sets in the plane of unit diameter,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li>
<li>the maximum inclusion problem,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li>and the <a href="/facts/Moser%27s_worm_problem/nq47sn6j">Moser's worm problem</a> for a convex universal cover of minimal size for the collection of planar curves of unit length.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>A. B. Ivanov (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Blaschke_selection_theorem">"Blaschke selection theorem"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>V. A. Zalgaller (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Metric_space_of_convex_sets">"Metric space of convex sets"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Kai-Seng Chou; Xi-Ping Zhu (2001). <i>The Curve Shortening Problem</i>. CRC Press. p. 45. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-58488-213-1.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Paul J. Kelly; Max L. Weiss (1979). Geometry and Convexity: A Study in Mathematical Methods. Wiley. pp. Section 6.4. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Paul J. Kelly; Max L. Weiss (1979). Geometry and Convexity: A Study in Mathematical Methods. Wiley. pp. Section 6.4. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Paul J. Kelly; Max L. Weiss (1979). Geometry and Convexity: A Study in Mathematical Methods. Wiley. pp. Section 6.4. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Wetzel, John E. (July 2005). "The Classical Worm Problem --- A Status Report". Geombinatorics. 15 (1): 34–42. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Blaschke selection theorem open-in-new

Blaschke selection theorem