On the conjecture by Demyanov-Ryabova in converting finite exhausters

TL;DR

This paper proves the Demyanov-Ryabova conjecture regarding cycle lengths in converting exhausters in affinely independent settings and offers a combinatorial reformulation to aid future research.

Contribution

The paper confirms the conjecture for affinely independent cases and introduces a combinatorial reformulation of the problem.

Findings

Cycle length in converting exhausters is at most two in affinely independent cases.

A combinatorial reformulation of the conjecture is provided.

The conjecture is proven for a specific geometric setting.

Abstract

In this paper, we prove the conjecture of Demyanov and Ryabova on the length of cycles in converting exhausters in an affinely independent setting and obtain a combinatorial reformulation of the conjecture. Given a finite collection of polyhedra, we can obtain its "dual" collection by forming another collection of polyhedra, which are obtained as the convex hull of all support faces of all polyhedra for a given direction in space. If we keep applying this process, we will eventually cycle due to the finiteness of the problem. Demyanov and Ryabova claim that this cycle will eventually reach a length of at most two. We prove that the conjecture is true in the special case, that is, when we have affinely independent number of vertices in the given space. We also obtain an equivalent combinatorial reformulation for the problem, which should advance insight for the future work on this problem.