Research

My research interests are convex geometry and related problems in geometric (functional) analysis. In many cases, the underlying motivation is to find functional analogs of known concepts from the classical theory of convex bodies. In particular, I have worked on valuations of convex functions, especially on classification results and their integral geometry. I am furthermore interested in volume inequalities and their functional counterparts, as well as Minkowski-type problems. Other topics of my research include isometries and supports of (mixed) area measures.

Grants/Fellowships/Awards

12/2022–11/2025: FWF stand-alone project, Integral geometry on convex functions (Project Number P 36210)

Project Webpage | Due to my parental leave, the project lead was temporarily transferred to Monika Ludwig from 03/2023 to 09/2023.

10/2020–02/2023: Erwin Schrödinger Fellowship of the FWF, Hessian inequalities and extensions to Sobolev spaces (Project Number J 4490)

Project Webpage | A short article about my experiences abroad was published here.

10/2020: Best Paper Award 2019 of the Faculty of Mathematics and Geoinformation (TU Wien)

PhD students

12/2022–10/2025: Mohamed Abdeldjalil Mouamine

Thesis: On Vector-Valued Valuations on Convex Functions

Conference Organization

07/2025: Focused Workshop on Current Trends in Geometric Valuation Theory, Erdős Center, Budapest, Hungary

jointly with Jan Kotrbatý

04/2022: Mini-Symposium on Geometric Probability and Valuation Theory, TU Wien, Austria

jointly with Florian Besau

Publications

Preprints

- Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries | arXiv:2508.11600

jointly with Jacopo Ulivelli

- The vectorial Hadwiger theorem on convex functions | arXiv:2504.04952

jointly with Mohamed Abdeldjalil Mouamine

- A Klain-Schneider theorem for vector-valued valuations on convex functions | arXiv:2503.07287

jointly with Mohamed Abdeldjalil Mouamine

- Inequalities and counterexamples for functional intrinsic volumes and beyond | arXiv:2412.05001

jointly with Jacopo Ulivelli

- Kubota-type formulas and supports of mixed measures | arXiv:2401.16371

jointly with Daniel Hug and Jacopo Ulivelli

Published/Accepted Original Research

Additive kinematic formulas for convex functions | arXiv:2403.06697

Canadian Journal of Mathematics, to appear.

jointly with Daniel Hug and Jacopo Ulivelli

The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas | arXiv:2109.09434

American Journal of Mathematics 147 (2025), no. 4, 927–955.

jointly with Andrea Colesanti and Monika Ludwig

The Hadwiger theorem on convex functions, I | arXiv:2009.03702

Geometric and Functional Analysis 34 (2024), no. 6, 1839–1898.

jointly with Andrea Colesanti and Monika Ludwig

The Hadwiger theorem on convex functions, IV: The Klain approach | arXiv:2201.11565

Advances in Mathematics 413 (2023), Art. 108832, 35 pp.

jointly with Andrea Colesanti and Monika Ludwig

Metrics and isometries for convex functions | arXiv:2010.12846

International Mathematics Research Notices 2022, no. 18, 14496–14563.

jointly with Ben Li

The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures | arXiv:2111.05648

Calculus of Variations and Partial Differential Equations 61 (2022), no. 5, Art. 181, 37 pp.

jointly with Andrea Colesanti and Monika Ludwig

Valuations on log-concave functions | arXiv:1707.06428

Journal of Geometric Analysis 31 (2021), no. 6, 6427–6451.

SL(n) invariant valuations on super-coercive convex functions | arXiv:1903.04225

Canadian Journal of Mathematics 73 (2021), no. 1, 108–130.

- A homogeneous decomposition theorem for valuations on convex functions | arXiv:1908.10742

Journal of Functional Analysis 279 (2020), no. 5, Art. 108573, 25 pp.

jointly with Andrea Colesanti and Monika Ludwig

- Hessian valuations | arXiv:1711.09632

Indiana University Mathematics Journal 69 (2020), no. 4, 1275–1315.

jointly with Andrea Colesanti and Monika Ludwig

- Volume, polar volume and Euler characteristic for convex functions | arXiv:1806.11084

Advances in Mathematics 344 (2019), 340–373.

- Valuations on convex functions | arXiv:1703.06455

International Mathematics Research Notices 2019, no. 8, 2384–2410.

jointly with Andrea Colesanti and Monika Ludwig

- Minkowski valuations on convex functions | arXiv:1707.05242

Calculus of Variations and Partial Differential Equations 56 (2017), no. 6, Art. 162, 29 pp.

jointly with Andrea Colesanti and Monika Ludwig

Book Chapters

Valuations on convex bodies and functions | arXiv:2302.00416

In: Convex Geometry: Cetraro, Italy 2021, Lecture Notes in Mathematics 2332, CIME Foundation subseries, pp. 19–78, Springer, Cham, 2023.

jointly with Monika Ludwig

Miscellaneous

Vector-valued valuations on convex bodies and convex functions

In: Convex Geometry and its Applications, Oberwolfach Reports 21 (2024), no. 4, 3311–3313

Characterizations of intrinsic volumes on convex bodies and convex functions

Snapshots of modern mathematics from Oberwolfach 11/2022, 12 pp.

Singular Hessian valuations and mixed Monge-Ampère measures

In: Convex Geometry and its Applications, Oberwolfach Reports 18 (2021), no. 4, 3220–3222

Volume, polar volume and Euler characteristic for convex functions

In: Convex Geometry and its Applications, Oberwolfach Reports 15 (2018), no. 4, 3243–3244

Note that arXiv preprints may differ from the final published versions. A complete list of my arXiv uploads can be found here: https://arxiv.org/a/mussnig_f_1.html

Conferences/Workshops/Activities

2025

11/2025 Convex Geometric Analysis and Stochastic Geometry, Tianyuan Mathematics Research Center, Kunming, China (upcoming)

07/2025 Focused Workshop on Current Trends in Geometric Valuation Theory, Erdős Center, Budapest, Hungary

05/2025 2025 Szeged Workshop on Convexity, Szeged, Hungary

- - - - Talk (20 min.): A Pólya–Szegő-type inequality for convex functions

02/2025 Fifth Austrian Day of Women in Mathematics, TU Wien, Austria

2024

12/2024 MFO Workshop Convex Geometry and its Applications, Oberwolfach, Germany

- - - - Talk (30 min.): Vector-valued valuations on convex bodies and convex functions

06/2024 Convex Geometry in Florence - 2024, Florence, Italy

- - - - Talk (20 min.): An extremal property of rotation epi-means

05/2024 Conference on Convex Geometry and Related PDEs, Changsha, China

- - - - Invited Talk (45 min.): Integral geometry and supports of mixed measures

02/2024 Fourth Austrian Day of Women in Mathematics, RICAM, Linz, Austria

02/2024 Research Visit in Jena, Germany – Thomas Wannerer, Georg Hofstätter

- - - - Invited Talk (60 min.): Kubota-type formulas and supports of mixed measures

2023

11/2023 Privatissimum, Research Unit for Convex and Discrete Geometry, TU Wien, Austria

- - - - Talk (55 min.): On mixed measures, part I

09/2023 Conference on Convex Geometry and Geometric Probability, Paris Lodron University Salzburg, Austria

- - - - Talk (20 min.): Inequalities and counterexamples for functional intrinsic volumes

09/2023 ÖMG Meeting 2023, Mini symposium Probabilistic Methods in Convexity, Graz, Austria

- - - - Invited Talk (25 min.): Integral geometry and supports of mixed measures

Due to a personal emergency, I had to cancel this talk.

06/2023 INdAM Meeting: Convex Geometry - Analytic Aspects, Cortona, Italy

- - - - Plenary Talk (50 min.): An elementary question about functional intrinsic volumes

06/2023 Geometric Valuation Theory - from convex sets to functions, Castro Urdiales, Spain

- - - - Plenary Lecture (55 min.) : Integral geometry on convex functions

03/2023 Seminar, Research Unit for Variational Analysis, Dynamics & Operations Research, TU Wien, Austria

- - - - Talk (60 min.): Integral geometry on convex functions

02/2023 Third Austrian Day of Women in Mathematics (online)

02/2023 1 Week Research Visit in Karlsruhe, Germany – Daniel Hug

- - - - Talk (60 min.): Integral geometry on convex functions

2022

09/2022 Harmonic Analysis Methods in Geometric Tomography, ICERM, Providence, Rhode Island, USA

- - - - Invited Talk (45 min.): Functional intrinsic volumes | Video

05/2022 2022 Szeged Workshop on Convexity, Szeged, Hungary

- - - - Plenary Talk (50 min.): Functional intrinsic volumes

05/2022 CMO Workshop Integral and Metric Geometry, Oaxaca, Mexico (online)

04/2022 Mini-Symposium on Geometric Probability and Valuation Theory, Vienna, Austria

02/2022 Second Austrian Day of Women in Mathematics (online)

2021

12/2021 MFO Workshop Convex Geometry and its Applications, Oberwolfach, Germany

- - - - Talk (20 min.): Singular Hessian valuations and mixed Monge-Ampère measures

10/2021 Privatissimum, Research Unit for Convex and Discrete Geometry, TU Wien, Austria

- - - - Talk (50 min.): Singular Hessian valuations and mixed Monge-Ampère measures

09/2021 DMV-ÖMG Annual Conference 2021, Section Geometry and Topology, University of Passau, Germany (online)

- - - - Talk (25 min.): Intrinsic volumes on convex functions

09/2021 Convex, Integral and Stochastic Geometry - International Conference in Honour of Rolf Schneider and Wolfgang Weil, Bad Herrenalb, Germany

- - - - Talk (30 min.): Integral geometric formulas for functional intrinsic volumes

08/2021 CIME Summer School on Convex Geometry, Cetraro, Italy

- - - - Lecture Series: Valuations on convex bodies and convex functions (jointly with Monika Ludwig)

07/2021 2-Day Research Visit in Vienna, Austria – Monika Ludwig

06/2021 CMS 75th+1 Anniversary Summer Meeting, Session on "New perspectives on the Brunn-Minkowski theory", Ottawa, Canada (online)

- - - - Invited Talk (30 min.): Functional intrinsic volumes and Hadwiger's theorem for convex functions

05/2021 Privatissimum, Research Unit for Convex and Discrete Geometry, TU Wien, Austria (online)

- - - - Talk (50 min.): Metrics and isometries for convex functions

03/2021 Workshop: High dimensional measures: geometric and probabilistic aspects, HIM, Bonn, Germany (online)

03/2021 Conference Current Trends in Convex Geometry, Bremen, Germany and Murcia, Spain (online)

- - - - Poster: Metrics and isometries for convex functions

02/2021 First Austrian Day of Women in Mathematics (online)

01/2021 Winter School on “The Interplay between High-Dimensional Geometry and Probability”, HIM, Bonn, Germany (online)

2020

12/2020 Privatissimum, Research Unit for Convex and Discrete Geometry, TU Wien, Austria (online)

- - - - Talk (60 min.): Hadwiger's theorem for convex functions

01/2020 2-Day Research Visit in Vienna, Austria – Monika Ludwig

2019

11/2019 AGA Seminar, Tel Aviv University, Israel

- - - - Talk (60 min.): A McMullen-type decomposition for valuations on convex functions

09/2019 Conference on Convex, Discrete and Integral Geometry, Jena, Germany

- - - - Talk (25 min.): SL(n) invariant valuations on convex functions

07/2019 4-Day Research Visit in Frankfurt, Germany – Andreas Bernig

- - - - Talk (60 min.): SL(n) invariant valuations on convex functions

04/2019 3 Day Research Visit in Florence, Italy – Andrea Colesanti

2018

12/2018 MFO Workshop Convex Geometry and its Applications, Oberwolfach, Germany

- - - - Talk (20 min.): Volume, polar volume and Euler characteristic for convex functions

07/2018 Block Course on Floating Bodies and Affine Invariants by Elisabeth Werner, Frankfurt, Germany

04/2018 Privatissimum, Research Unit for Convex and Discrete Geometry, TU Wien, Austria

- - - - Talk (60 min.): Volume, polar volume and Euler characteristic for convex functions

03/2018 3 Day Research Visit in Florence, Italy – Andrea Colesanti

2017

06/2017 Convex Discrete and Integral Geometry, Bedlewo, Poland

- - - - Talk (30 min.): Valuations on log-concave functions

2016

10/2016 INdAM Workshop Analytic Aspects of Convexity, Rome, Italy

- - - - Talk (25 min.): Valuations on convex functions

07/2016 Conference on Convex and Discrete Geometry on the Occassion of Peter M. Gruber's 75th birthday, Vienna, Austria

05/2016 Perspectives on Integral Geometry, Athens, Georgia, USA

- - - - Talk (30 min.): Minkowski valuations on convex functions

02/2016 4 Week Research Visit in Florence, Italy – Andrea Colesanti

- - - - Talk (60 min.): Minkowski valuations on log-concave functions

2015

12/2015 Privatissimum, Research Unit for Convex and Discrete Geometry, TU Wien, Austria

- - - - Talk (60 min.): Minkowski valuations on log-concave functions

09/2015 CMO Workshop Affine Geometric Analysis, Oaxaca, Mexico

07/2015 BMS Summer School 2015: Convex Geometry - Discrete and Computational, Berlin, Germany

06/2015 Workshop on Integral Geometry and Valuation Theory, Zurich, Switzerland

2014

09/2014 Workshop on Tensor Valuations in Stochastic Geometry and Imaging, Sønderborg, Denmark