Lightness functions
type: Item
instance of: scholarly article
title: Lightness functions
zbMATH Open document ID: 1261.52002
author: Stefano Campi
author: Richard J. Gardner
author: Paolo Gronchi
author: Markus Kiderlen
published in: Advances in Mathematics
publication date: 2012-11-20T00:00:00Z
review text: 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 \i…
reviewed by: Vladimir Golubyatnikov
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52B05
zbMATH DE Number: 6108923
zbMATH Keywords: convex bodies
zbMATH Keywords: brightness function
zbMATH Keywords: lightness function
zbMATH Keywords: illumination
zbMATH Keywords: Lambert's cosine law
zbMATH Keywords: shadow boundary
zbMATH Keywords: geometric tomography
zbMATH Keywords: surface area
zbMATH Keywords: Blaschke sum
zbMATH Keywords: Blaschke indecomposable bodies
MaRDI profile type: Publication
full work available at URL: j.aim.2012.08.015
OpenAlex ID: W4206148521
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DOI: 10.1016/J.AIM.2012.08.015
description: scientific article; zbMATH DE number 6108923