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 on convex and log-concave functions - especially on classification results. Furthermore, I am interested in volume inequalities and their functional counterparts.
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)
Conference Organization
04/2022: Mini-Symposium on Geometric Probability and Valuation Theory at TU Wien, Austria
jointly with Florian Besau
Publications
Preprints
Additive kinematic formulas for convex functions | arXiv:2403.06697
jointly with Daniel Hug and Jacopo Ulivelli
Kubota-type formulas and supports of mixed measures | arXiv:2401.16371
jointly with Daniel Hug and Jacopo Ulivelli
The Hadwiger theorem on convex functions, I | arXiv:2009.03702
jointly with Andrea Colesanti and Monika Ludwig
Published/Accepted Original Research
The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas | arXiv:2109.09434
American Journal of Mathematics, to appear.
jointly with Andrea Colesanti and Monika Ludwig
Advances in Mathematics 413 (2023), Art. 108832, 35 pp.
jointly with Andrea Colesanti and Monika Ludwig
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
Journal of Geometric Analysis 31 (2021), no. 6, 6427–6451
Canadian Journal of Mathematics 73 (2021), no. 1, 108–130
Journal of Functional Analysis 279 (2020), no. 5, Art. 108573, 25 pp.
jointly with Andrea Colesanti and Monika Ludwig
Indiana University Mathematics Journal 69 (2020), no. 4, 1275–1315
jointly with Andrea Colesanti and Monika Ludwig
Advances in Mathematics 344 (2019), 340–373
International Mathematics Research Notices 2019, no. 8, 2384–2410
jointly with Andrea Colesanti and Monika Ludwig
Calculus of Variations and Partial Differential Equations 56 (2017), no. 6, Art. 162, 29 pp.
jointly with Andrea Colesanti and Monika Ludwig
Book Chapters
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
Snapshots of modern mathematics from Oberwolfach 11/2022, 12 pp.
In: Convex Geometry and its Applications, Oberwolfach Reports 18 (2021), no. 4, 3220–3222
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
2024
02/2024 Fourth Austrian Day of Women in Mathematics, RICAM, Linz, Austria
02/2024 Research Visit in Jena, Germany – Thomas Wannerer, Georg Hofstätter
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
Invited 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)
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