A Minkowski-type inequality for convex surfaces in the hyperbolic 3-space

TL;DR

This paper establishes a new Minkowski-type inequality for convex surfaces in hyperbolic 3-space by analyzing the normal flow and applying isoperimetric principles, also providing elementary proofs for classical inequalities in Euclidean and spherical spaces.

Contribution

It introduces a novel Minkowski-type inequality specific to hyperbolic 3-space and offers elementary proofs for classical inequalities in Euclidean and spherical geometries.

Findings

Derived a new inequality for hyperbolic convex surfaces

Computed surface areas via normal flow

Provided elementary proofs for classical Minkowski inequalities

Abstract

In this note we derive a new Minkowski-type inequality for closed convex surfaces in the hyperbolic 3-space. The inequality is obtained by explicitly computing the area of the family of surfaces obtained from the normal flow and then applying the isoperimetric inequality. Using the same method, we also we give elementary proofs of the classical Minkowski inequalities for closed convex surfaces in the Euclidean 3-space and in the 3-sphere.