# Symmetry from sectional integrals for convex domains

## TL;DR

This paper proves that certain integral geometric conditions on a convex domain imply it must be a ball, and characterizes the associated functions as radial, extending symmetry results in convex geometry.

## Contribution

It establishes a new characterization of spherical symmetry for convex domains based on sectional integrals and Radon transforms.

## Key findings

- Convex domain with specific Radon transform properties must be a ball.
- The associated function is uniquely radial about the domain's center.
- Provides a new geometric criterion for symmetry in convex analysis.

## Abstract

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ ($n \ge 2$). In this work, we prove that if there exists an integrable function $f$ such that it's Radon transform over $(n-1)$-dimensional hyperplanes intersecting the domain $\Omega$ is a strictly positive function of distance to the nearest parallel supporting hyperplane to $\Omega$, then $\Omega$ is a ball and the function $f$ is a unique radial function about the centre of $\Omega$.