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We consider the convexity properties of distortion functionals, particularlythe linear distortion, defined for homeomorphisms of domains in Euclidean$n$-spaces, $n\geq 3$. The inner and outer distortion functionals are lowersemi-continuous in all dimensions and so for the curve modulus or analyticdefinitions of quasiconformality it ifollows that if $ \{ f_{n}\}_{n=1}^{\infty} $ is a sequence of $K$-quasiconformal mappings (here $K$depends on the particular distortion functional but is the same for everyelement of the sequence) which converges locally uniformly to a mapping $f$,then this limit function is also $K$-quasiconformal.Despite a widespread beliefthat this was also true for the geometric definition of quasiconformality(defined through the linear distortion $H({f_{n}})$), T. Iwaniec gave aspecific and surprising example to show that the linear distortion functionalis not always lower semicontinuous on uniformly converging sequences ofquasiconformal mappings. Here we show that this failure of lower semicontinuityis common, perhaps generic in the sense that under mild restrictions on aquasiconformal $f$, there is a sequence $ \{f_{n} \}_{n=1}^{\infty} $ with ${f_{n}}\to {f}$ locally uniformly and with $\limsup_{n\to\infty} H( {f_{n}})<H({f})$. Our main result shows this is true for affine mappings. Addressingconjectures of F.W. Gehring and Iwaniec we show the jump up in the limit can bearbitrarily large and give conjecturally sharp bounds : for each$\alpha<\sqrt{2}$ there is ${f_{n}}\to {f}$ locally uniformly with $f$ affineand \[ \alpha \; \limsup_{n\to\infty} H( {f_{n}}) < H( {f}) \] We conjecture$\sqrt{2}$ to be best possible.

The Generic Failure of Lower-semicontinuity for the Linear Distortion Functional