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Online seminar “The isoperimetric inequality outside convex sets”

The online semi­nar "The isop­eri­met­ric ine­quali­ty out­side con­vex sets" origi­nated from the idea to offer inter­ested stu­dents from dif­ferent uni­versi­ties an in­sight into an ad­vanced math­emat­ical topic. To this pur­pose, Prof. Nico­la Fusco was invit­ed, and intro­duced vari­ous con­cepts of geo­met­ric measure theo­ry dur­ing eight Zoom lec­tures.

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.

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Text: Marwin Forster, Christian Parsch, Elite Graduate Program „TopMath”