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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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