A Centroid for Sections of a Cube in a Function Space, with application to Colorimetry

TL;DR

This paper extends the concept of centroids to certain subsets of function spaces, specifically in $L^1[0,1]$, using finite-dimensional subspaces, with applications to spectral reflectance estimation in colorimetry.

Contribution

It introduces a novel approach to defining centroids in function spaces via finite-dimensional subspaces, overcoming measure invariance issues, with practical applications in colorimetry.

Findings

Successful definition of centroids for intersections of cubes with affine subspaces in $L^1[0,1]$

Application of Laplace Transform and saddlepoint methods for asymptotic analysis

Demonstrated relevance to spectral reflectance estimation in colorimetry

Abstract

The definition of the centroid in finite dimensions does not apply in a function space because of the lack of a translation invariant measure. Another approach, suggested by Nik Weaver, is to use a suitable collection of finite-dimensional subspaces. For a specific collection of subspaces of $L^1[0,1]$, this approach is shown to be successful when the subset is the intersection of a cube with a closed affine subspace of finite codimension. The techniques used are the classical Laplace Transform and saddlepoint method for asymptotics. Applications to spectral reflectance estimation in colorimetry are presented.