# Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities

## TL;DR

This paper extends the parallel coordinates method to three dimensions and proves a conjecture that the ball maximizes the first Robin eigenvalue among convex domains with equal surface area, with partial results in higher dimensions.

## Contribution

It introduces a three-dimensional extension of parallel coordinates and proves a conjecture about Robin eigenvalues for convex domains, with partial higher-dimensional results.

## Key findings

- The ball maximizes the first Robin eigenvalue among convex domains with equal surface area.
- Extension of parallel coordinates method to three dimensions.
- Partial results for eigenvalue maximization in arbitrary dimensions.

## Abstract

In this paper we show how the method of parallel coordinates can be extended to three dimensions. As an application, we prove the conjecture of Antunes, Freitas and Krej\v{c}i\v{r}\'ik \cite{AFK} that "the ball maximises the first Robin eigenvalue with negative boundary parameter among all convex domains of equal surface area" under the weaker restriction that the boundary of the domain is diffeomorphic to the sphere and convex or axiconvex. We also provide partial results in arbitrary dimensions.