3. Geometric problems
Questions involving geometric quantities such as isoperimetric profile, moment of inertia, rearrangements.-
Cheeger’s inequality on convex sets
Problem 3.1.
[Ilias Ftouhi] For an open set of finite measure $\Omega\subset\mathbb{R}^n$, denote $\lambda_1(\Omega)$ the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and $h(\Omega)$ the Cheeger constant which may be defined as \[h(\Omega)=\inf_{A\subset \Omega}\frac{|\partial A|}{|A|}=\inf_{u\in\mathcal{C}^\infty_c(\Omega)}\frac{\int_{\Omega}|\nabla u|}{\int_{\Omega}u}\] Cheeger’s inequality is \[\lambda_1(\Omega)\geq \frac{1}{4}h(\Omega)^2\] and the constant $\frac{1}{4}$ is sharp in the sense that the ratio $\frac{\lambda_1(\Omega)}{h(\Omega)^2}$ tends to $\frac{1}{4}$ when $\Omega$ is taken to be the unit ball of $\mathbb{R}^n$ for $n\to +\infty$. It is however not sharp when $\Omega$ is restricted to be a planar convex sets, as is shown for instance by the result of I. Ftouhi: \[\frac{\lambda_1(\Omega)}{h(\Omega)^2}\geq 0.902\] Ilias Ftouhi’s conjecture is that the optimal bound is $\frac{2\pi^2}{(2+\sqrt{\pi})^2}\approx 1.387...$, which is reached exactly when $\Omega$ is a square.
An optimal set is known to exist, by standard compactness arguments, but it is unknown whether this optimal set is a polygon. A first stepping stone would be to examine the case of polygonal sets, where the Cheeger set (the optimal set “$A$” in the definition) should coincide with the boundary of $\Omega$ on some part, or leave the boundary tangentially following circle arcs (as described in [kawohl, lachand-robert]). -
Slicing problem in 3 dimensions
Problem 3.2.
[Luis Rademacher] Suppose $\Omega \subset \mathbb{R}^3$ is convex, normalized to have volume $1$, and with its center of mass at the origin. Let \[ A_{ij} = \int_\Omega x_i x_j \, dx . \] The ball minimizes $\det A$. The problem is to find the shape that maximizes $\det A$.
This functional is affine invariant under linear maps with determinant $1$ (volume preserving). The scale invariant form of the functional is $\det A/|\Omega|^{n+2}$, where here $n=3$.
In two dimensions, the maximizer is the triangle. In dimensions $n \geq 3$, the problem has been open since 1986: one possibility raised by Luis Rademacher is that it is reached by a simplex in dimension $n=3$. -
Rearrangement problem
Problem 3.3.
[Paul Simanjuntak] Let $F:\mathbb{R}\to\mathbb{R}$ be an even convex function, then it is known (from Xi, Zhao (2022)) that for any function $f,g\in L^1(\mathbb{R}^n)$ (for some $n\geq 1$), denoting $f^*,g^*$ the symmetrically decreasing rearrangement of $f,g$, we have \[\iint_{R^n\times\mathbb{R}^n}F(x\cdot y)f(x)g(y)dxdy\geq \iint_{R^n\times\mathbb{R}^n}F(x\cdot y)f^*(x)g^*(y)dxdy\] However it is unclear whether the convexity constraint on $F$ is sharp. The two questions raised by Paul Simanjuntak are as follows:
1) Can the hypothesis on $F$ be weakened ?
2) Does this inequality extend to other settings, for instance Gaussian measures ?
Cite this as: AimPL: Symmetry-breaking of optimal shapes, available at http://aimpl.org/symmetrybreaking.