Let
$\lambda \geqq 1$,
$E={E}^{N}$ and
$g$ be continuous
on
$E\times E\times E$ with
$g\left(a,\cdot ,\cdot \right)$ convex,
$g\left(a,kb,kc\right)={k}^{2}g\left(a,b,c\right)$ for all
real
$k$ and
$\left({b}^{2}+{c}^{2}\right)\u2215\lambda \leqq g\left(a,b,c\right)\leqq \lambda \left({b}^{2}+{c}^{2}\right)$ for all
$a,b,c\in E$ where
${b}^{2}=\parallel b{\parallel}^{2}$. If
$f\left(a,d\wedge e\right)=\underset{b\wedge c=d\wedge e}{min}g\left(a,b,c\right)$ then
$f$
is a permissible integrand for the twodimensional parametric variational
problem.
Let
$\gamma $ be a simple
closed curve in
$E$,
$B$ be the closed unit circle
in the plane,
$C$ be the
collection of functions
$x$
continuous on
$B$
into
$E$ for
which
$x\partial B\in \gamma $ and
$D=\left\{x\in Cx\text{isa}D\text{map}\right\}$. Suppose
that
$D$ is
not empty. It was shown in ‘A problem of least area’, [7], that the problem of minimizing
$I\left(f\right)$ over
$D$ is equivalent
to minimizing
$I\left(g\right)$
over
$D$
where
$I\left(f,x\right)=\iint \phantom{\rule{0.3em}{0ex}}f\left(x,p\wedge q\right),I\left(g,x\right)=\iint \phantom{\rule{0.3em}{0ex}}g\left(x,p,q\right),p={x}_{u}$,
$q={x}_{v}$ and both integrals
are taken over
$B$. The
minimizing solution of
$I\left(g\right)$
is known to have differentiability properties corresponding to
$g$, and this solution
also minimizes
$I\left(f\right)$.
The function
$f$ is simple,
that is, for each
$a\in E$, each
supporting linear functional to
$f\left(a,\cdot \right)$
is simple. If
$N=3$,
then, of course, each parametric integrand is simple. In this
paper we show that for each simple parametric integrand
$F$ there exists
$G$, satisfying the conditions
imposed upon
$g$,
such that
$F$ is
obtained from
$G$
as
$f$ was
obtained from
$g$.
