A short note on the Erdős-Debrunner inequality (Q2470902)

The inequality referred to in the title [see \textit{C. Schütz}, Elem. Math. 61, No. 4, 155--172 (2006; Zbl 1128.51007) for its history] states that ``If a triangle \(XYZ\) is inscribed in a triangle \(ABC\) -- with \(X, Y, Z\) on the sides \(BC, CA,\) and \(AB\) -- then \(\sigma(XYZ)\geq \min (\sigma(BXZ), \sigma(CXY), \sigma(AYZ))\) with equality if and only if \(X, Y\), and \(Z\) are the midpoints of the sides \(BC, CA,\) and \(AB\),'' where \(\sigma(MNP)\) stands for the area of the triangle \(MNP\). Noting that, if one denotes by \(M_p(x,y,z)\) the mean \((\frac{x^p+y^p+z^p}{3})^{1/p}\) for \(p\neq 0\), \(\root 3\of{xyz}\) for \(p=0\), and \(\min(x,y,z)\) for \(p=-\infty\), then one would have \(\lim_{p\rightarrow -\infty} M_p(x,y,z)=M_{-\infty}(x,y,z)\), so one expects inequalities of the type \(\sigma(XYZ)\geq M_p(\sigma(BXZ), \sigma(CXY), \sigma(AYZ))\) to hold for \(p>-\infty\) as well, and if they do, then one would like to know the maximum value among all such \(p\), denoted by \(p_{\max}\). The author proves that \(-1\leq p_{\max}\leq -\frac{\ln(3/2)}{\ln(2)}\), and suspects that \(p_{\max}=-\frac{\ln(3/2)}{\ln(2)}\).