The volume enclosed by an n-dimensional Lam\'e curve

TL;DR

This paper calculates the volume of the region enclosed by an n-dimensional Lamé curve, which is defined by a specific sum of powers of coordinates equaling a constant, expanding understanding of high-dimensional geometric shapes.

Contribution

The paper provides a formula for the volume of the body enclosed by the n-dimensional Lamé curve, a problem not previously addressed in high-dimensional geometry.

Findings

Derived an explicit formula for the volume of the Lamé curve body.

Extended geometric analysis to n-dimensional spaces.

Contributed to the mathematical understanding of high-dimensional shapes.

Abstract

We compute the volume of the body enclosed by the $n$-dimensional Lam\'e curve defined by $\sum_{i=1}^nx_i^b= E$.