Orthogonal Projection of Convex Sets with a Differentiable Boundary

TL;DR

This paper explores the relationship between the derivatives of the Minkowski functional of a convex set with a smooth boundary and the orthogonal projection boundary onto subspaces, deriving equations and illustrating with a normed ball example.

Contribution

It establishes a topological connection between Minkowski functional derivatives and projection boundaries, providing a new system of equations for orthogonal projections.

Findings

Derived a system of equations for orthogonal projections.

Linked derivatives of Minkowski functional to projection boundaries.

Illustrated results with a normed ball projection example.

Abstract

Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior) and the boundary of its orthogonal projection onto the linear subspaces of the Euclidean space. A system of equations for these orthogonal projections is derived from this topological link. This result is illustrated by the projection of the unit ball of norm $4$ in $\mathbb{R}^3$ on a plane.