J Optim Theory Appl (2017) 174:712–727
On the Conjecture by Demyanov–Ryabova
in Converting Finite Exhausters
Received: 5 February 2017 / Accepted: 11 July 2017 / Published online: 18 July 2017
© Springer Science+Business Media, LLC 2017
Abstract The Demyanov–Ryabova conjecture is a geometric problem originating
from duality relations between nonconvex objects. Given a ﬁnite collection of poly-
topes, one obtains its dual collection as convex hulls of the maximal facet of sets in
the original collection, for each direction in the space (thus constructing upper convex
representations of positively homogeneous functions from lower ones and, vice versa,
via Minkowski duality). It is conjectured that an iterative application of this conver-
sion procedure to ﬁnite families of polytopes results in a cycle of length at most two.
We prove a special case of the conjecture assuming an afﬁne independence condition
on the vertices of polytopes in the collection. We also obtain a purely combinatorial
reformulation of the conjecture.
Keywords Demyanov–Ryabova · Exhausters · Polytope · Convex hull · Afﬁnely
independent · Combinatorial reformulation
Mathematics Subject Classiﬁcation 90C27 · 52B11
Given a ﬁnite collection of polytopes, we can obtain its dual by taking convex hulls
of the support faces for every nonzero direction. Then, if we continue this process,
Communicated by Aris Daniilidis.
Discipline of Mathematical Sciences, School of Science, RMIT University, Melbourne, VIC 3000,