  In this paper, we revisit the notion of length measures associated to planar
closed curves. These are a special case of area measures of hypersurfaces which
were introduced early on in the field of convex geometry. The length measure of
a curve is a measure on the circle $\mathbb{S}^1$ that intuitively represents
the length of the portion of curve which tangent vector points in a certain
direction. While a planar closed curve is not characterized by its length
measure, the fundamental Minkowski-Fenchel-Jessen theorem states that length
measures fully characterize convex curves modulo translations, making it a
particularly useful tool in the study of geometric properties of convex
objects. The present work, that was initially motivated by problems in shape
analysis, introduces length measures for the general class of Lipschitz
immersed and oriented planar closed curves, and derives some of the basic
properties of the length measure map on this class of curves. We then focus
specifically on the case of convex shapes and present several new results.
First, we prove an isoperimetric characterization of the unique convex curve
associated to some length measure given by the Minkowski-Fenchel-Jessen
theorem, namely that it maximizes the signed area among all the curves sharing
the same length measure. Second, we address the problem of constructing a
distance with associated geodesic paths between convex planar curves. For that
purpose, we introduce and study a new distance on the space of length measures
that corresponds to a constrained variant of the Wasserstein metric of optimal
transport, from which we can induce a distance between convex curves. We also
propose a primal-dual algorithm to numerically compute those distances and
geodesics, and show a few simple simulations to illustrate the approach.
