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The derivation of counterexamples to L¹ estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn’s inequality, and of the corresponding geometrically nonlinear rigidity result, in L¹. Secondly, we construct a function f:ℝ²→ℝ which is separately convex but whose gradient is not in BV_loc, in the sense that the mixed derivative ∂²f/∂x₁∂x₂ is not a bounded measure.