Some functions on the set of triangles or quadrangles (Q788281)

Let \(M=(ABCD)\) be a quadrangle with area S(M) on a Euclidean 2-space \(E^ 2\) and let \(Q=(x,y,z)\) be a point of \(E^ 3\). We assume that \(E^ 2\) is defind by \(z=0\) in \(E^ 3\). Define a function \(g_ M(x,y,z)\) by \(g_ M(Q)=| \Delta QAB|^ 2+| \Delta QBC|^ 2+| \Delta QCD|^ 2+| \Delta QDA|^ 2\), and put \(G(M)=\int \exp(-4r^ 2g_ M)dE^ 3\). Then, the main result is stated as follows: \[ G(M)\leq(\pi^{3/2}/4r^ 3S(M)^{3/2})\cdot \exp(-r^ 2S(M)^ 2), \] where equality holds if and only if M is a square. This is an example of trials of calculating the areas of minimal surfaces with given boundaries; quadrangles in \(E^ 3\).