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
- 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
- LAM NGUYEN, Memorial University of Newfoundland
- 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
- BEATRICE-HELEN VRITSIOU, University of Alberta
- ELISABETH WERNER, Case Western Reserve University
- JIE XIAO, Memorial University
$C^1$-maximizer of $p$-mean torsion rigidity on convex bodies [PDF]
- LAM NGUYEN, Memorial University of Newfoundland
-
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