Mixed Volumes. Minkowski Problem. Selected Global Problems in Geometric Partial Differential Equations

Abstract

Let Vⁿ⁺¹ be a (n + 1)-dimensional Euclidean vector space. We denote by Eⁿ⁺¹ the (n + 1)-dimensional Euclidean point space associated with Vⁿ⁺¹ (see §1). Then every vector a ∈ Vⁿ⁺¹ generates a parallel translation

$$ {p_{a}}:{E^{{n + 1}}} \to {E^{{n + 1}}}$$

(7.1)

in the following way: for any point X ∈ Eⁿ⁺¹

$$ {p_{a}}\left( X \right) = X', $$

, where X’ is the terminal point of the vector $\overline {XX'} = a$.