# Harmonic Analysis and Affine Isoperimetric Inequalities in Convex Geometry

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Case Western Reserve University (OH) · $147,128

## Abstract

This project focuses on two types of mathematical problems: tomographic problems and isoperimetric problems. Tomography concerns the retrieval of information about a geometric object based on limited information arising from its cross-sections or shadows (projections). For example, think of trying to determine the volume of a mountain based on the size of its shadows at different times of the day. Isoperimetric problems arise in geometry and optimization and have been of interest for thousands of years, dating back to ancient Greece. In a modern context, isoperimetric problems can involve studying the regularity of high-dimensional information. In complex data sets, high dimensionality often results in a regularizing effect.  Tomographic and isoperimetric problems have been highly influential in many scientific disciplines, including physics, engineering, and computer science.  Beyond their historical significance and applicability, these problems appeal to a wide audience because they are often intuitively stated and explained, while their solutions are difficult and require sophisticated mathematical techniques.


The principal investigator of this project seeks to address problems arising naturally from geometry and harmonic analysis by employing techniques involving the Fourier transform, Radon transform, and other tools from various mathematical fields. Among these proposed problems are new affine invariant estimates for mixed-Sobolev norms, estimates for Radon and Co

## Key facts

- **NSF award ID:** 2452384
- **Awardee organization:** Case Western Reserve University (OH)
- **SAM.gov UEI:** HJMKEF7EJW69
- **PI:** Michael Roysdon
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $147,128
- **Funds obligated:** $147,128
- **Transaction type:** Standard Grant
- **Period:** 06/15/2025 → 09/30/2025

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2452384

## Citation

> US National Science Foundation, Award 2452384, Harmonic Analysis and Affine Isoperimetric Inequalities in Convex Geometry. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2452384. Licensed CC0.

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