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Internal cubic symmetric forms in a small number of variables. (English) Zbl 1135.26022
Let \(f\) be a real symmetric cubic form in \(n\) variables. The authors wish to characterise the case that \(\mu=\root3\of f\) is a mean: let us say that \(\mu\) is internal if \(\min(x_1,\dots,x_n)\leq\mu(x_1,\dots,x_n)\leq\max(x_1,\dots,x_n)\) for all nonnegative \(x_1,\dots,x_n\). This note proves that if \(n\leq4\), then it suffices to test internality for \(x_1,\dots,x_n\) in \(\{0,1\}\). The case \(n\geq5\) is open.

MSC:
26E60 Means
26D05 Inequalities for trigonometric functions and polynomials
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