In the theory of convex subsets in a Euclidean space, an important role is played by Minkowski duality (the polar transform of a convex set, or the Legendre transform of a convex set). We consider conformally flat Riemannian metrics on the n-dimensional unit sphere and their embeddings into the isotropic cone of the Lorentz space. For a given class of metrics, we define and carry out a detailed study of the Legendre transform.

中文翻译：

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保形度量的极坐标变换

在欧几里得空间中的凸子集理论中，Minkowski对偶性（凸集的极坐标变换或凸集的Legendre变换）起着重要作用。我们考虑n维单位球面上的保形黎曼度量，并将其嵌入到Lorentz空间的各向同性锥中。对于给定的度量标准，我们定义并进行了Legendre变换的详细研究。