2025 CMS Winter Meeting
Toronto, Dec 5 - 8, 2025
Recent progress in convex and discrete geometry
Org: Ferenc Fodor (University of Szeged, Hungary and University of Calgary, Canada) and Alina Stancu (Concordia University, Canada)
[PDF]
Org: Ferenc Fodor (University of Szeged, Hungary and University of Calgary, Canada) and Alina Stancu (Concordia University, Canada)
[PDF]
- GERGELY AMBRUS, University of Szeged, Hungary
- KAROLY BEZDEK, University of Calgary
- TED BISZTRICZKY, University Calgary
- DMITRY FAIFMAN, University of Montreal
- ALEX IOSEVICH, University of Rochester
- JASKARAN KAIRE, University of Manitoba
- PAVLOS KALANTZOPOULOS, University of Waterloo
- DYLAN LANGHARST, Carnegie Mellon University
Grünbaum’s inequality for probability measures [PDF]
- KAROLY BEZDEK, University of Calgary
-
Given a convex body $K$ in $\mathbb{R}^n$, a natural question is: if one partitions the body into two pieces along its barycenter, how small can each piece be? By “partition along its barycenter”, we mean intersecting $K$ with a half-space whose boundary is a hyperplane containing said barycenter. Grünbaum showed that the volume of each piece is at least $\left(\frac{n}{n+1}\right)^n$ times the total volume of K. Furthermore, this constant is sharp: there is equality if and only if $K$ is a cone, which means there exists a $(n - 1)$-dimensional convex body $L$ and a vector $b$, such that $K$ has face $L$ and vertex $b$ (i.e. $K$ is the convex hull of $b$ and $L$).
\
In this work, which is joint with M. Fradelizi, J. Liu, F. Marin Sola, and S. Tang, we are interested in generalizing Grünbaum’s inequality to other measures. Our main results are a sharp inequality for the Gaussian measure and a sharp inequality for s-concave probability measures. The characterization of the equality case is of particular interest. Along the way, we discover new facts about the equality case of the Borell-Brascamp-Lieb inequality.
- SERGII MYROSHNYCHENKO, University of the Fraser Valley
Polytope Reconstruction: floating and illuminating structures [PDF]
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The surface of buoyancy (or surface of centers) of a convex body plays a central role in the study of floating bodies. Its properties and significance have been the focus of extensive recent work by D. Florentin, H. Huang, D. Ryabogin, C. Schutt, B. Slomka, E. Werner, B. Zawalski, N. Zhang, among others. In this talk, we address the problem of unique determination of a convex polytope by its surface of buoyancy or by its Dupin floating body. We also consider the dual questions in the setting of illumination bodies. The presented results are based on joint work with S. Dann and O. Herscovici.
- LAM NGUYEN, Memorial University of Newfoundland
Logarithmic Sobolev, Poincaré, and Beckner Inequalities on Hyperbolic Spaces [PDF]
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This talk presents recent progress in the study of logarithmic Sobolev, Poincaré, and Beckner inequalities on hyperbolic spaces. We focus on determining the best constants and exploring their close connection to Gaussian measures. We also discuss new versions of Beckner inequalities that come from studying heat flow on hyperbolic spaces. This work is a collaboration with Anh Do, Guozhen Lu, and Debdip Ganguly.
- DEBORAH OLIVEROS, UNAM Queretaro, Mexico
- DMITRY RYABOGIN, Kent State University
- EGON SCHULTE, Northeastern University
- CARSTEN SCHÜTT, University of Kiel, Germany
- KATERYNA TATARKO, University of Waterloo
- VIKTOR VIGH, University of Szeged, Hungary
- ELISABETH WERNER, Case Western Reserve University
- JIE XIAO, Memorial University
$C^1$-maximizer of $p$-mean torsion rigidity on convex bodies [PDF]
- DMITRY RYABOGIN, Kent State University
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Given a bounded domain $B\subset\mathbb R^{n\ge 2}$ with its boundary $\partial B$, a solution $u_B$ of the torsion problem
$$\begin{cases}\Delta u_B=-1 &\text{in}\ \ B;\ u_B=0&\text{on}\ \ \partial B,
\end{cases}
$$
is called a stress function of $B$. Via the torsion rigidity
$$
\int_{B}|\nabla u_B(x)|^2\,dx,
$$
this talk is about to show that the maximization problem for $[1,\infty)\ni p$-mean torsion rigidity
$$(\star)\ \
\underset{\text{all convex bodies $B\subset\mathbb R^n$}}{\sup}\int_{{B}}\Bigg(\frac{|\nabla u_B(x)|^2}{{|{B}|^\frac2n}}\Bigg)^{p}\frac{dx}{|{B}|},
$$
is achievable and the boundary $\partial B_\bullet$ of any maximizer $B_\bullet$ of $(\star)$ is $C^1$-smooth, thereby finding that if $|\nabla u_{B_\bullet}|$ is constant on ${\partial B_\bullet}$ then $B_\bullet$ is a Euclidean ball.
- BARTLOMIEJ ZAWALSKI, Case Western Reserve University