Let $\lambda_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $\lambda_1$ satisfies a Brunn--Minkowski inequality: $K \mapsto \lambda_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j \geq 2$ and $N \geq 2$, if $K \mapsto (f \circ \lambda_j)(K)$ is concave on the family of convex bodies in $\mathbb{R}^N$ for some function $f: (0, \infty) \to \mathbb{R}$, then $f$ must be constant.
For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $\mu$ on $\mathbb R^n$, all nonempty symmetric convex sets $K,L\subseteq\mathbb R^n$, and all $\lambda\in[0,1]$, $$ \mu(\lambda K+(1-\lambda)L)^{p_n} \ge \lambda\mu(K)^{p_n}+(1-\lambda)\mu(L)^{p_n}...
We prove that for every fixed $\lambda>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-\lambda n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erd\H{o}s 973 can exist. The proof combines a trun...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codime...
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp exam...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
We prove an analogue of Gr\"unbaum's inequality on the sphere. Let $n \geq 3$ and let $K$ be a convex body on $\mathbb S^{n-1}\subset \mathbb R^n$ with centroid at $\theta\in \mathbb S^{n-1}$. Then for any $u\in \mathbb S^{n-1}$ that is orthogonal to $\theta$ we have $$\sigma(K\cap u^+) \ge \left(1-\frac{1}{n}\right)^{...
S. Myroshnychenko, D. Ryabogin, Kateryna Tatarko et al.· 0 citations
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