We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies, and Alexandrov–Fenchel inequality is interpreted as the Lorentzian reversed Cauchy–Schwarz inequality. We deduce that the space of similarity classes of convex bodies has a proper geodesic distance with curvature bounded from below by $-1$ (in the sense of Alexandrov). In dimension 3, this space is homeomorphic to the space of distances with non-negative curvature on the 2-sphere, and this latter space contains the space of flat metrics on the 2-sphere considered by W. P. Thurston. Both Thurston’s and the area distances rely on the area form. So the latter may be considered as a generalization of the “real part” of Thurston’s construction.

中文翻译：

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凸体形状的双曲几何

我们使用本征面积来定义$n$维欧几里得空间中凸体同位类空间上的距离，使其与无限维双曲空间的凸子集等距。环境洛伦兹结构是凸体本征面积形式的扩展，亚历山德罗夫-芬切尔不等式被解释为洛伦兹逆柯西-施瓦茨不等式。我们推断凸体的相似类空间具有适当的测地线距离，曲率从下方以$-1$为界（在亚历山德罗夫的意义上）。在第 3 维中，该空间同胚于 2 球体上具有非负曲率的距离空间，而后者空间包含 WP Thurston 所考虑的 2 球体上的平面度量空间。瑟斯顿距离和区域距离都依赖于区域形式。所以后者可以被认为是对瑟斯顿构造的“实部”的概括。