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Normal bundles of convex bodies. (English) Zbl 1296.52003
Let \( C \subset {\mathbb E}^n \) be a convex body symmetric with respect to the origin, and \({\mathrm {bd}} C \) be its boundary. The author defines its normal bundle cone \( {\mathcal N}_C \subset {\mathbb E}^{n^2} \) as the positive hull of the set \( \{x \otimes n_C (x); \, x \in {\mathrm {bd}} C \}\), and its polar cone \( {\mathcal N}_C^* \) as \( \{ Y \in {\mathrm {lin}} \,{\mathcal N}_C : \, X \cdot Y \geq 0, \, X \in {\mathcal N}_C \} \). Here \({\mathrm {lin}} \) denotes the linear hull and \( n_C(x) \) is the exterior normal vector to \({\mathrm {bd}} C \) at \( x\).
The main results of the paper are devoted to:
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Detailed description of general properties of the normal bundle cone of a convex set, in particular relations of the type \({\mathcal N}_{C + D} \subseteq {\mathcal N}_C + {\mathcal N}_D \) (Theorem 1);
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Description of the space of normal bundle cones as the subset of all convex cones in \( {\mathbb E}^{n^2} \) (Theorem 2);
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Estimates of dimensions of the normal bundle cones, in particular \( \frac{1}{2} d(d+1) \leq { \mathrm {dim}} {\mathcal N}_C \leq d^2 \). Studies of sharpness of these estimates (Theorem 3);
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Polarity and transposition properties of the normal bundle cones, such as the relations of the type \({\mathcal N}_C \subseteq {\mathcal N}_C^* \) and analysis of their sharpness (Theorem 4).
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Theorem 5. There is one-to-one correspondence between the family of all rank one faces of the cone \( {\mathcal N}_C \) and the family of all planar shadow boundaries of the body \( C \) with respect to parallel illumination.
The author formulates three natural conjectures concerning dimensions of the normal bundle cones; relations of isometries (and linear automorphisms) of \( {\mathcal N}_C \) to those of \( C \); and possibility of generalizations of the Theorem 5 to higher codimensions.

MSC:
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
53A05 Surfaces in Euclidean and related spaces
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