# On convex bodies that are characterizable by volume function

## TL;DR

This survey reviews the history and key open problems related to volume functions like the convex-hull, covariogram, and brightness functions in three-dimensional convex bodies, highlighting their mathematical significance.

## Contribution

It provides a comprehensive overview of the main concepts, results, and open problems concerning volume functions characterizable by convex bodies in 3D.

## Key findings

- Identification of key volume functions in convex geometry
- Summary of major open problems and conjectures
- Highlighting the importance of computational methods in recent research

## Abstract

The "old-new" concept of convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems connected with these functions whose serious investigation closed before the "age of computers". In this survey, we concentrate only on the three-dimensional case, we will mention the most important concepts, statements, and problems.