  The equioscillation theorem interleaves the Haar condition, the existence and
uniqueness and strong uniqueness of the optimal Chebyshev approximation and its
characterization by the equioscillation condition in a way that cannot extend
to multivariate approximation: Rice~[\emph{Transaction of the AMS}, 1963] says
''A form of alternation is still present for functions of several variables.
However, there is apparently no simple method of distinguishing between the
alternation of a best approximation and the alternation of other approximating
functions. This is due to the fact that there is no natural ordering of the
critical points.'' In addition, in the context of multivariate approximation
the Haar condition is typically not satisfied and strong uniqueness may hold or
not. The present paper proposes an multivariate equioscillation theorem, which
includes such a simple alternation condition on error extrema, existence and a
sufficient condition for strong uniqueness. To this end, the relationship
between the properties interleaved in the univariate equioscillation theorem is
clarified: first, a weak Haar condition is proposed, which simply implies
existence. Second, the equioscillation condition is shown to be equivalent to
the optimality condition of convex optimization, hence characterizing
optimality independently from uniqueness. It is reformulated as the
synchronized oscillations between the error extrema and the components of a
related Haar matrix kernel vector, in a way that applies to multivariate
approximation. Third, an additional requirement on the involved Haar matrix and
its kernel vector, called strong equioscillation, is proved to be sufficient
for the strong uniqueness of the solution. These three disconnected conditions
give rise to a multivariate equioscillation theorem, where existence,
characterization by equioscillation and strong uniqueness are separated,
without involving the too restrictive Haar condition. Remarkably, relying on
optimality condition of convex optimization gives rise to a quite simple proof.
Instances of multivariate problems with strongly unique, non-strong but unique
and non-unique solutions are presented to illustrate the scope of the theorem.
