Hessian inequalities and extensions to Sobolev spaces
Hessian inequalities and extensions to Sobolev spaces
Disciplines
Mathematics (100%)
Keywords
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Convex Geometry,
Brunn-Minkowski theory,
Inequality,
Hessian measure,
Convex Function,
Sobolev space
Among all geometric figures in the plane, the circle has the property that it has the largest possible area for a given circumference. Similarly, in three-dimensional space, a sphere has the largest volume for a given surface area. In mathematics, this fact is generally expressed by the so-called isoperimetric inequality, which is a special case of even more general volume inequalities. Recently, the concept of surface area and similar related characteristics has been extended from geometric figures or bodies to convex functions. This is known as Hessian valuations. The aim of the project is to generalize the volume inequalities which are known for geometric objects to the new Hessian valuations. Furthermore, both the Hessian valuations as well as the new inequalities are to be extended to an even larger class of functions. One anticipates that this will not only lead to new insights and applications for functions themselves, but also to new approaches to problems related to the classical volume inequalities.
Volume and surface area are natural quantities that one can assign to a given shape to measure its size. These quantities are generalized by the intrinsic volumes which are central objects in convex and integral geometry, where so-called convex bodies are studied. Recently, functional intrinsic volumes on convex functions were introduced. These operators associate a number to a given convex function and serve as functional analogs of the intrinsic volumes. The premise of this project is based on the fact that the classical intrinsic volumes satisfy some well-known significant inequalities, such as the isoperimetric inequality. It is therefore only natural to ask whether similar inequalities also hold for their new functional counterparts. While it turned out that such conjectured inequalities do not hold in the functional world, the outcomes of this project still provide many new exciting insights into functional intrinsic volumes. One major result of this project is a new Cauchy-Kubota type formula for convex functions which is a far-reaching generalization of a classical formula due to Cauchy that dates back to the 19th century. This formula not only allows for new representations of functional intrinsic volumes but also explains the possible singularities of the density functions that appear in their definitions. Next, a new functional version of the classical Steiner formula provides a new way to define functional intrinsic volumes as coefficients of a naturally arising polynomial and allows to describe these operators with the help of special mixed Monge-Ampre measures. Last but not least, the projects results can also explain why the initially conjectured inequalities do not hold true. Nonetheless, new Wulff-type inequalities for functional intrinsic volumes were found.
Research Output
- 14 Citations
- 8 Publications
- 3 Scientific Awards
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2023
Title Valuations on Convex Bodies and Functions DOI 10.48550/arxiv.2302.00416 Type Other Author Ludwig M -
2021
Title The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas DOI 10.48550/arxiv.2109.09434 Type Preprint Author Colesanti A -
2022
Title The Hadwiger theorem on convex functions, IV: The Klain approach DOI 10.48550/arxiv.2201.11565 Type Other Author Colesanti A -
2022
Title Characterizations of intrinsic volumes on convex bodies and convex functions Type Other Author Mussnig F Conference Snapshots of modern mathematics from Oberwolfach Pages 1-12 -
2022
Title Characterizations of intrinsic volumes on convex bodies and convex functions Type Other Author Mussnig F Conference Snapshots of modern mathematics from Oberwolfach Pages 1-12 -
2023
Title Valuations on Convex Bodies and Functions DOI 10.1007/978-3-031-37883-6_2 Type Book Chapter Author Ludwig M Publisher Springer Nature Pages 19-78 Link Publication -
2022
Title The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge–Ampère measures DOI 10.1007/s00526-022-02288-3 Type Journal Article Author Colesanti A Journal Calculus of Variations and Partial Differential Equations Pages 181 Link Publication -
2023
Title The Hadwiger theorem on convex functions, IV: The Klain approach DOI 10.1016/j.aim.2022.108832 Type Journal Article Author Colesanti A Journal Advances in Mathematics Pages 108832 Link Publication
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2022
Title Principal speaker at the 2022 Szeged Workshop on Convexity Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2023
Title Plenary Lecture at Geometric Valuation Theory - from convex sets to functions Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2022
Title Main Speaker at the INdAM Meeting "CONVEX GEOMETRY - ANALYTIC ASPECTS" Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International