In the real Hilbert space of self-adjoint elements of the tensor product ${\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}$, there are two natural cones besides the cone ${\mathfrak {P}}_{0}$ of positive semi-definite elements. The one is and the other is the cone ${\mathfrak {P}}_{-}$, dual to ${\mathfrak {P}}_{+}$ with respect to the inner product. Then, ${\mathfrak {P}}_{+} \subset {\mathfrak {P}}_{0} \subset {\mathfrak {P}}_{-}.$ A weak order relation $\geq $ is introduced by Our interest is in finding bounds for the ratio $|||T|||/|||S|||$ for $S \geq T \geq 0$, where $| | |\cdot| | |$ is one of the operator norm, the trace norm, and the Hilbert-Schmidt norm. $${\mathfrak{P}}_{+} := \text{the convex hull of}\ \{X\otimes Y ; ~ 0 \leq X \in {\mathbb{M}}_{m},~ 0 \leq Y \in {\mathbb{M}}_{n}\}$$ $$\mathbf{S} \succeq \mathbf{T}\quad \overset{def}{\Longleftrightarrow}\quad \mathbf{S} - \mathbf{T} \in {\mathfrak{P}}_{-}.$$