  Non-locality is being intensively studied in various PDE-contexts and in
variational problems. The numerical approximation also looks challenging, as
well as the application of these models to Continuum Mechanics and Image
Analysis, among other areas. Even though there is a growing body of deep and
fundamental knowledge about non-locality, for variational principles there are
still very basic questions that have not been addressed so far. Taking some of
these as a motivation, we describe a general perspective on distinct classes of
non-local variational principles setting a program for the analysis of this
kind of problems. We start such program with the simplest problem possible:
that of scalar, uni-dimensional cases, under a particular class of
non-locality. Even in this simple initial scenario, one finds quite unexpected
facts to the point that our intuition about local, classic problems can no
longer guide us for these new problems. There are three main issues worth
highlighting, in the particular situation treated:
  $\bullet$ despite the different nature of non-locality as compared to the
classic, local situation, the natural underlying spaces still are the standard
Sobolev spaces;
  $\bullet$ no convexity of integrands is required for existence of minimizers;
  $\bullet$ optimality is formulated in terms of quite special integral
equations rather than differential equations.
  We are thus able to provide some specific answers to the initial questions
that motivated our investigation. In subsequent papers, we will move on to
consider the higher dimensional situation driven by the possibility that no
convexity or quasiconvexity might be involved in weak lower semicontinuity in a
full vector, higher dimensional situation.
