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Séminaire Lotharingien de Combinatoire, 78B.27 (2017), 12 pp.

# Jules Depersin, Stéphane Gaubert
and Michael Joswig

# A Tropical Isoperimetric Inequality

**Abstract.**
We introduce tropical analogues of the notion of volume of polytopes,
leading to a tropical version of the (discrete) classical
isoperimetric inequality. The planar case is elementary, but a
higher-dimensional generalization leads to an interesting class of
ordinary convex polytopes, characterizing the equality case in the
isoperimetric inequality. This study is motivated by open complexity
questions concerning linear optimization and its tropical analogs.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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