From Lagrange's identity $$ |\mathbf{a}|^2 |\mathbf{b}|^2 = (\mathbf {a \cdot b})^2 + |\mathbf {a \times b}|^2 $$ the Cauchy-Schwarz inequality follows $$ |\mathbf{a}|^2 |\mathbf{b}|^2 \geq (\mathbf {a \cdot b})^2 $$ However one could also derive the following inequality $$ |\mathbf{a}|^2 |\mathbf{b}|^2 \geq |\mathbf {a \times b}|^2 $$ which is based on the "antisymmetric" part of Lagrange's identity. Does this last inequality have a name? Is it used somewhere?
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If we are talking about dimension $3$, notice that the interpretation is very simple: $$|\bf a\times b|\le |a|\cdot|b|$$ says that the volume of the parallelogram defined by $\bf a$ and $\bf b$ is at most the product of the lengths of the sides.
Probably because of this it would be a bit too much to give it a name, even more when it is only a very particular case of the inequality $$ \lvert\det A|\le\prod_{i=1}^n\|v_i\| $$ if $v_1,\ldots,v_n$ are the columns of $A$.
John B
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