Product cones in dense pairs

LetM = ⟨ M , < , + , ⋯ ⟩ $\mathcal {M}=\langle M, <, +, \dots \rangle$ be an o‐minimal expansion of an ordered group, andP ⊆ M $P\subseteq M$ a dense set such that certain tameness conditions hold. We introduce the notion of a product cone inM ∼ = ⟨ M , P ⟩ $\widetilde{\mathcal {M}}=\langle \mathcal {M}, P\rangle$ , and prove: if M $\mathcal {M}$ expands a real closed field, thenM ∼ $\widetilde{\mathcal {M}}$ admits a product cone decomposition. If M $\mathcal {M}$ is linear, then it does not. In particular, we settle a question from [10].