Lightness functions
Let \({\mathcal K}^n\) be the class of convex compact bodies in \({\mathbb R}^n\), and \(\delta (\cdot,\cdot)\) be the Hausdorff distance. Given a convex body \(K \subset {\mathbb R}^n\) and vectors \(v, w \in {S}^{n-1}\), the authors define the \textit{brightness function} \(R_K(v,w)\) which is the area of the projection orthogonal to \(v\) of the part of the surface of \(K\) that is both illuminated by a light source in the direction \(w\) and visible from the direction \(v\), and the class of \textit{lightness functions} \(Q_K(v,w)\) which depend on some function \(f_{w}= h_{w} 1_{w^+}\) and generalize the \textit{Lambert's cosine law}. Here \(w^+ = \{ x\in {\mathbb R}^n; \, x \cdot w > 0\}\), \(1_{w^+}\) is its indicator function, and \(h_w\) is continuous and strictly positive on \(S^{n-1} \cap w^{\bot}\). Let \(h_K\) be the support function of \(K\) and \(K(u)=\{x \in K; \, x \cdot u = h_K (u) \}\) be the support set of \(K\) with outer unit normal \(u\). The \textit{shadow boundary} \(G_w (K) = \bigcup_{u \in S^{n-1}\cap w^{\bot}} K(u)\) is called \textit{sharp} if \(K(u)\) is a singleton for all \(u \in w^{\bot}\). The unit ball \(B^n \subset {\mathbb R}^n\) is called \textit{essentially illuminated} by finite collection of directions (light sources) \(w_1,\dots ,w_k \in S^{n-1}\) if \(S^{n-1}\) is the closure of all boundary points of \(B^n\) that are illuminated by at least one of these light sources. The main results of the paper are: Theorem 4.2. Let \(K, K' \in {\mathcal K}^n\) be centrally symmetric, and their shadow boundaries in some direction \(w \in S^{n-1}\) be sharp. If \(Q_K (v,w) = Q_{K'} (v,w)\) for all \(v \in S^{n-1}\), then \(K=K'\), up to translation. Theorem 4.4. Let \( K, K' \in {\mathcal K}^n\) be strictly convex. If \(w_1, \dots , w_k \in S^{n-1}\) essentially illuminate \(B^n\), and \(Q_K (v, w_i) = Q_{K'} (v,w_i)\) for all \(i=1,\dots,k\) and all \(v \in S^{n-1}\), then \(K=K'\), up to translation. Theorem 4.5. Let \(w_1,\dots,w_k \in S^{n-1}\), and let \(T\) be the part of \(B^n\) illuminated from these directions. The following statements are equivalent: \((i)\) If \(K, K' \in {\mathcal K}^n\) and \(Q_K (v, w_i) = Q_{K'} (v,w_i)\) for all \(i=1,\dots ,k\) and \(v \in S^{n-1}\), then \(K = K'\), up to translations. \((ii)\) The set \(S^{n-1} \setminus T\) is empty or a singleton. Theorem 5.1. Let \(w \in S^{n-1}\) and let \( 0<r<R\). If \( n=2\), there is a constant \(c=c(R)\) such that \(\delta (K,K') \leq c \| R_K(\cdot, w) - R_{K'} (\cdot, w)\|^{2/3}_2 \) for all o-symmetric strictly convex bodies \( K \) and \(K'\) with \( rB^2 \subset K\), \( K' \subset RB^2\). If \( n \geq 3\) and \( \gamma > 0\), there is a constant \(c=c(\gamma, r, R)\) such that \( \delta (K,K') \leq c\| R_K(\cdot,w) - R_{K'}(\cdot,w)\|^{2/(n(n+1) - \gamma}_2\) for all o-symmetric strictly bodies \( K \) and \(K'\) with \( rB^n \subset K \), \( K' \subset RB^n\). Here a body is called \textit{o}-symmetric if it is centrally symmetric, with center at the origin. An analogous stability version of Theorem 4.5 is given by Theorem 5.2. Let \({\mathcal Q}(K_0, U) = \{ K \in {\mathcal K}^n ; \, Q_K (v_i, w_i) = Q_{K_0} (v_i, w_i),\, i = 1, \dots ,k \}\). Here \(U: = \{ (v_i, w_i), \, i = 1, \dots, k \}\). The authors study properties of bodies in \({\mathcal Q}(K_0, U)\) of maximal and minimal volume. The next condition is assumed to be satisfied in what follows. Condition (38) Let \( U \) be as above, and let for each \(u \in S^{n-1} \) there exists \( i \in \{1,\dots ,k \}\) such that \( u \) is simultaneously illuminated from the direction \( w_i \) and visible when looking in the direction \( v_i\). Theorem 6.4. Let \( K_0 \in {\mathcal K}^n\), \( U \) be as above. Assume that the functions \( f_{w_i}\) are continuous. Then there is a convex body in \( {\mathcal Q}(K_0, U) \), unique up to translation, of maximal volume. The Theorem 6.5. states similar result for \({\mathcal Q}^{-}(K_0, U) = \{ K \in {\mathcal K}^n ; \, Q_K (v_i, w_i) \leq Q_{K_0} (v_i, w_i),\, i = 1, \dots ,k \}\). If \( U \) is as above, then the hyperplanes \( v_i^{\bot} \), \( w_i^{\bot} \) decompose \( S^{n-1} \) to a finite collection of spherical convex polyhedra. The vertices of these polyhedra are called \( U\)-\textit{nodes}. Theorem 6.6. Let \( K_0\) and \( U \) be as above. If \( Q_K = R_K \) and \( K^+ \) is a convex body of maximal volume in \( {\mathcal Q}(K_0, U) \), then \( K^+ \) is a polytope with each of its facets orthogonal to one of its \( U \)-nodes. Analogous results on bodies in \( {\mathcal Q}(K_0, U)\) of minimal volumes, and of maximal (or minimal) surface area are given by the Theorems 6.12, 6.14, 7.1, 7.2. The upper bounds for the number of the facets of corresponding minimal (or maximal) volume (or surface area) polytopes are obtained there as well.
scientific article; zbMATH DE number 6108923
scientific article