A 2000 year old geometric problem
Amongst all geometric shapes with a given perimeter, which one has the largest surface area? This question may have already concerned the ancient queen Dido, who according to legend, was allowed to claim as much land as she could enclose by a band of fixed length. It was already clear to the ancient Greeks that the optimal shape is the circle. However, Dido was possibly allowed to use a straight piece of coastline “for free” as part of her boundary and in this case a semicircle bordering the coast would have been optimal.
Could an even better solution be found if the coastline wasn’t straight? This is indeed the case: For example, a peninsula of any arbitrary size could be enclosed as long as the connection to the land was narrow enough.
However, a semicircle around a straight piece of coast is optimal if there aren’t any peninsulas. In mathematics, in such a case, where the connecting line segment of any two points in the water always lies again completely in the water, the water surface is said to be convex. This optimality result is the so-called isoperimetric problem outside convex regions, the proof of which was studied by the students of the Elite Graduate Program "TopMath" in the seminar.
Solved with modern mathematical methods
Mathematically, this problem has recently been solved in two papers that use methods from geometric measure theory. This advanced mathematical subfield combines concepts of analysis and geometry. Prof. Nicola Fusco from Università degli Studi di Napoli (Italy) is an expert in this field and co-authored one of the papers. He presented some results in this online seminar, which was attended by interested master's and doctoral students in mathematics from several universities in Germany and abroad.
During the eight lectures, Prof. Fusco gave deep insights into the field of geometric measure theory. The participants were taught how basic geometric concepts such as perimeter, angle, surface area and curvature can be defined and applied in this very abstract world.
Further informations:
https://www.ma.tum.de/en/news-events/studies-information/topmath/archive/onlineseminar-fusco.html
Text: Marwin Forster, Christian Parsch, Elite Graduate Program „TopMath”