A way from the isoperimetric inequality in the plane to a Hilbert space

TL;DR

This paper generalizes the classical isoperimetric inequality to a broader setting and uses it to construct a Hilbert space and a reproducing kernel Hilbert space, linking geometric inequalities to functional analysis.

Contribution

It introduces a novel generalization of the isoperimetric inequality and demonstrates its application in constructing a Hilbert space and an RHKS from geometric principles.

Findings

Established a generalized isoperimetric inequality in the plane.

Constructed a Hilbert space using the generalized inequality.

Connected geometric inequalities with the structure of reproducing kernel Hilbert spaces.

Abstract

We will formulate and prove a generalization of the isoperimetric inequality in the plane. Using this inequality we will construct an unitary space - and in consequence - an isomorphic copy of a separable infinite dimensional Hilbert space, which will appear also an RHKS.