  Effective periods were defined by Kontsevich and Zagier as complex numbers
whose real and imaginary parts are values of absolutely convergent integrals of
$\mathbb{Q}$-rational functions over $\mathbb{Q}$-semi-algebraic domains in
$\mathbb{R}^d$. The Kontsevich-Zagier period conjecture states that any two
different integral expressions of a period are related by a finite sequence of
transformations only using three rules respecting the rationality of functions
and domains: integral addition by integrands or domains, change of variables
and Stokes' formula.
  In this paper, we introduce two geometric interpretations of this conjecture,
seen as a generalization of Hilbert's third problem involving either compact
semi-algebraic sets or rational polyhedra equipped with piece-wise algebraic
forms. Based on known partial results for analogous Hilbert's third problems,
we study possible geometric schemes to prove this conjecture and their
potential obstructions.
