On the importance of the connectedness assumption in the statement of the optimal $p$-compliance problem

TL;DR

This paper emphasizes the critical role of the connectedness assumption in the optimal p-compliance problem, demonstrating its necessity for the existence of solutions in length-penalized and constrained formulations.

Contribution

It provides a proof highlighting the essential nature of the connectedness assumption for solution existence in the optimal p-compliance problem.

Findings

Connectedness assumption is crucial for solution existence.

Proof demonstrates the necessity of connectedness in problem formulation.

Highlights limitations when the assumption is not met.

Abstract

In this short note we provide a proof of the importance of the connectedness assumption in the statement of the optimal $p$-compliance problem with length penalization and in the statement of the constrained form of this problem for the existence of solutions.