= Triweight density minimizes integrated squared curvature
{title2=$R(h'')\geq R(f_0'')$}
Let $f_0$ be the <triweight kernel> and h a twice continuously differentiable density with matching zero mean and variance. Put $e=h-f_0$ and $c=35/32$. Two integrations by parts over its support give $\int e''f_0''=48c[h(1)+h(-1)]+c\int_{|x|>1}(360x^2-72)h(x)dx\geq0$, using the zero mass and second moment of e. Therefore $R(h'')=R(f_0'')+R(e'')+2\int e''f_0''\geq R(f_0'')$. The endpoint terms matter because the third derivative jumps there.
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