On the local minimizers of the Mahler volume

Details On the local minimizers of the Mahler volume On the local minimizers of the Mahler volume

2011-04-19

arxiv.org/abs/1104.3663v1

Mathematics

Optimization and Control

Scientific paper

We focus on the analysis of local minimizers of the Mahler volume, that is to say the local solutions to the problem $$\min\{M(K):=|K||K^\circ|\;/\;K\subset\R^d\textrm{open and convex}, K=-K\}, $$ where $K^\circ:=\{\xi\in\R^d ; \forall x\in K, x\cdot\xi<1\}$ is the polar body of $K$, and $|\cdot|$ denotes the volume in $\R^d$. According to a famous conjecture of Mahler the cube is expected to be a global minimizer for this problem. We express the Mahler volume in terms of the support functional of the convex body, which allows us to compute first and second derivatives, and leads to a concavity property of the functional. As a consequence, we prove first that any local minimizer has a Gauss curvature that vanishes at any point where it is defined. Going more deeply into the analysis in the two-dimensional case, we also prove that any local minimizer must be a parallelogram. We thereby retrieve and improve an original result of Mahler, who showed that parallelograms are global minimizers in dimension 2, and also the case of equality of Reisner, who proved that they are the only global minimizers.

Affiliated with

Also associated with

No associations

Rating

On the local minimizers of the Mahler volume does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the local minimizers of the Mahler volume, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the local minimizers of the Mahler volume will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-182525