6 problems

Thin-shell conjecture. The thin-shell constant is universally bounded:

KLS conjecture. For every isotropic logconcave density in $n$-dimensional Euclidean space, $\psi_n$ is bounded by a universal constant independent of $n$.

Let $u_0$ be a bounded nonnegative function in $\mathbf{R}^N$ with compact support. For each $1\leq\alpha<2$, Theorem 3.4 asserts that there exists $T_\alpha>0$ such that…

Let $\Omega\subset\mathbf{R}^N$ be a bounded convex domain, and let $\phi$ be the first positive Dirichlet eigenfunction of $-\Delta$ in $\Omega$. A function $u$ is 2-logconcave wh…

Let $\{d_{\ell}(m)\}$ be the sequence described above, and define the operator … A sequence is infinitely logconcave if $\mathfrak{L}^j(\{x_k\})$ is nonnegative for every…

Let $A_n(x)$ be the polynomial defined earlier in the paper, and let a sequence be infinite logconcave when every iterate of the operator $\mathfrak{L}$ produces a positive sequenc…