Let \(I\) be finite, \(w_i\gt 0\) for every \(i\), and \(p\gt 0\). The set \(\{ f\in \mathbb {R}^I:\sum _iw_i|f_i|^p=1\} \) is compact in the usual finite-dimensional topology.

Let \(I\) be finite, \(w_i\gt 0\), \(\sum _iw_i=1\), \(1\lt p\lt 2\), and \(0\le \rho \le 1\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). There exists a nonnegative vector \(g\) with \(G(g)=1\) and \(F(f)\le F(g)\) whenever \(G(f)=1\).

Let \(I\) be finite, let \(\varphi :\mathbb {R}\to \mathbb {R}\) be strictly concave on \([0,\infty )\), and let \(f:I\to [0,\infty )\) satisfy \(\varphi (f_i)=c\) for every \(i\) and some \(c\in \mathbb {R}\). There exist \(a,b\ge 0\) such that every \(f_i\) equals \(a\) or \(b\).

Let \(I\) be finite, \(w:I\to \mathbb {R}\), and \(p,\rho ,t\in \mathbb {R}\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). Then \(G(tf)=|t|^pG(f)\) and \(F(tf)=t^2F(f)\) for every real vector \(f\).

Let \(f,g:\mathbb {R}\to \mathbb {R}\) be differentiable at \(x\), with \(f^{\prime }(x)=a\), \(g^{\prime }(x)=b\), and \(g(x)=1\). If \(t\mapsto f(t)/g(t)^r\) has a local maximum at \(x\) for \(r\in \mathbb {R}\), then \(a=rf(x)b\).

Let \(I\) be finite, \(w:I\to \mathbb {R}\), \(p\ne 0\), and \(\rho \in \mathbb {R}\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). If \(G(f)\gt 0\) and \(c=G(f)^{-1/p}\), then \(G(cf)=1\) and \(F(cf)=F(f)/G(f)^{2/p}\).

Let \(I\) be finite, \(w:I\to \mathbb {R}\), \(p\gt 1\), \(\rho \in \mathbb {R}\), \(g_i\ge 0\), and \(h\in \mathbb {R}^I\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). At \(t=0\), the derivative of \(G(g+th)\) is \(p\sum _iw_ig_i^{p-1}h_i\), and that of \(F(g+th)\) is \(2[\rho ^2\sum _iw_ig_ih_i+(1-\rho ^2)(\sum _iw_ig_i)(\sum _iw_ih_i)]\).

Let \(I\) be finite, \(w_i\gt 0\), \(\sum _iw_i=1\), \(1\lt p\lt 2\), and \(0\le \rho \lt 1\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). If \(g_i\ge 0\), \(G(g)=1\), and \(F(f)\le F(g)\) whenever \(G(f)=1\), then \(F(g)\ge 1\) and \(F(g)g_i^{p-1}-\rho ^2g_i=(1-\rho ^2)\sum _jw_jg_j\) for every \(i\).

Let \(I\) be finite, \(w_i\gt 0\), \(\sum _iw_i=1\), \(1\lt p\lt 2\), and \(0\le \rho \lt 1\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). If \(g_i\ge 0\), \(G(g)=1\), and \(F(f)\le F(g)\) whenever \(G(f)=1\), then some \(a,b\ge 0\) satisfy \(g_i\in \{ a,b\} \) for every \(i\).