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.

在本文中，我们重新审视了与平面闭合曲线相关的长度测量概念。这些是早期在凸几何领域中引入的超曲面面积测量的特例。曲线的长度度量是对圆\(\mathbb {S}^1\)的度量直观地表示切线向量指向某个方向的曲线部分的长度。虽然平面闭合曲线不以其长度度量为特征，但基本的 Minkowski-Fenchel-Jessen 定理指出，长度度量完全表征凸曲线模平移，使其成为研究凸对象几何特性的特别有用的工具。目前的工作最初是由形状分析中的问题驱动的，介绍了一般类 Lipschitz 浸入和定向平面闭合曲线的长度测量，并推导出了此类曲线上长度测量图的一些基本属性。然后，我们专门关注凸形状的情况，并提出几个新结果。第一的，我们证明了与 Minkowski-Fenchel-Jessen 定理给出的某些长度度量相关的唯一凸曲线的等周特征，即它最大化共享相同长度度量的所有曲线之间的带符号面积。其次，我们解决了在凸平面曲线之间构建具有相关测地线路径的距离的问题。为此，我们引入并研究了长度度量空间上的新距离，该距离对应于最佳传输的 Wasserstein 度量的约束变体，我们可以从中得出凸曲线之间的距离。我们还提出了一种原始对偶算法来数值计算这些距离和测地线，并展示了一些简单的模拟来说明该方法。即它最大化共享相同长度度量的所有曲线之间的带符号面积。其次，我们解决了在凸平面曲线之间构建具有相关测地线路径的距离的问题。为此，我们引入并研究了长度度量空间上的新距离，该距离对应于最佳传输的 Wasserstein 度量的约束变体，我们可以从中得出凸曲线之间的距离。我们还提出了一种原始对偶算法来数值计算这些距离和测地线，并展示了一些简单的模拟来说明该方法。即它最大化共享相同长度度量的所有曲线之间的带符号面积。其次，我们解决了在凸平面曲线之间构建具有相关测地线路径的距离的问题。为此，我们引入并研究了长度度量空间上的新距离，该距离对应于最佳传输的 Wasserstein 度量的约束变体，我们可以从中得出凸曲线之间的距离。我们还提出了一种原始对偶算法来数值计算这些距离和测地线，并展示了一些简单的模拟来说明该方法。我们引入并研究了长度度量空间上的新距离，该距离对应于最佳传输的 Wasserstein 度量的约束变体，从中我们可以得出凸曲线之间的距离。我们还提出了一种原始对偶算法来数值计算这些距离和测地线，并展示了一些简单的模拟来说明该方法。我们引入并研究了长度度量空间上的新距离，该距离对应于最佳传输的 Wasserstein 度量的约束变体，从中我们可以得出凸曲线之间的距离。我们还提出了一种原始对偶算法来数值计算这些距离和测地线，并展示了一些简单的模拟来说明该方法。