Calc. Var. (2018) 57:60
https://doi.org/10.1007/s00526-018-1325-y
Calculus of Variations
A reciprocity principle for constrained isoperimetric
problems and existence of isoperimetric subregions in
convex sets
Michael Bildhauer
1
· Martin Fuchs
1
· Jan Müller
1
Received: 10 July 2017 / Accepted: 28 February 2018 / Published online: 13 March 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract It is a well known fact that in R
n
a subset of minimal perimeter L among all sets
of a given volume is also a set of maximal volume among all sets of the same perimeter L.
This is called the reciprocity principle for isoperimetric problems. The aim of this note is to
prove this relation in the case where the class of admissible sets is restricted to the subsets
of some subregion G R
n
. Furthermore, we give a characterization of those (unbounded)
convex subsets of R
2
in which the isoperimetric problem has a solution. The perimeter that
we consider is the one relative to R
n
.
Mathematics subject classification 49Q20
1 Introduction
In its classical form, the isoperimetric problem asks for the maximal area which can be
enclosed by a curve of given length. In modern mathematical terms, the task is to determine
a measurable subset of R
n
which has maximal Lebesgue measure A among all sets of a
given perimeter L ∈[0, ∞). Assuming the existence of a solution, Steiner in the first half
of the nineteenth century showed by means of elementary geometric arguments that in R
2
the only possible candidate for a solution is the circle of perimeter L (which has already
been suspected since antiquity). However, the existence part turns out to require a more
subtle reasoning. In nowadays mathematics, it is usually treated in the framework of the
Communicated by L. Ambrosio.
B
Michael Bildhauer
bibi@math.uni-sb.de
B
Martin Fuchs
fuchs@math.uni-sb.de
Jan Müller
jmueller@math.uni-sb.de
1
Department of Mathematics, Saarland University, P.O. Box 15 11 50, 66041 Saarbrücken, Germany
123
@Phil_Robichaud
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