For a convex body $K\subset\mathbb{R}^2$ that is symmetric with respect to the origin, and for a nonempty set $S\subset\mathbb{R}^2$, we study the $K$-Hausdorff distance from convex hull, defined by \begin{align*} d^{(K)}(S):=\sup_{x\in \text{conv}(S)}\inf_{s\in S}\|x-s\|_K, \end{align*} where $\|\cdot \|_K$ is the norm whose closed unit ball is $K$. We consider the problem of characterizing the origin symmetric convex bodies $K$ for which \begin{align*} d^{(K)}(A+B)^2\leq d^{(K)}(A)^2+d^{(K)}(B)^2 \end{align*} holds for all nonempty compact $A,B\subset\mathbb{R}^2$. We solve this problem, proving that this property holds if and only if $K$ is an ellipse centered at $0$. We then characterize the conditions for equality for this bound when $K$ is an ellipse.
Let $K\subset\mathbb{R}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(|K|/\omega_n)^{1/n}$. We consider the distribution function \[ D_K(\rho)=\mathbb P\{|X-Y|/R_K<\rho\}, \] and its homogeneous counterpart \begin{equation*} \mathcal J_\rho(K)=\iint_{K\times K}\mathbf{1}_{\{|x...
The symmetry measure of a convex body $S\subset\mathbb{R}^n$ is given by: $\mathrm{sym}(S):=\max\{\alpha\ge0:\text{ there exists }x\in S\text{ such that }-\alpha(S-x)\subseteq S-x\}$, where such an $x$ is called a Minkowski center. We prove that every convex body $S$ admits a $\sqrt{\frac{n}{\mathrm{sym}(S)}}$-rounding...
Godbersen's long-standing conjecture states that the mixed volumes of any convex body $K \subset \mathbb R^n$ satisfy $V(K[k], -K[n-k])\le \binom{n}{k} \operatorname{vol}(K)$ for any $k \in \{1, \dots, n - 1\}$, with equality if and only if $K$ is an $n$-simplex. Kotrbat\'y and Mouamine recently proved the inequality i...
O. Friedland, E. Putterman, Maud Szusterman· 0 citations
In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$\sigma_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overline{\Omega}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\Omega\Subset\R^n$, we establish ball rigidity results in al...
Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequentia...
Given $m \in \mathbb{N} \setminus \{0\}$ and $\rho > 0$, we find solutions
$(\lambda,u)$ to the problem
\begin{equation*}
\begin{cases}
\biggl(-\dfrac{d^2}{d x^2}\biggr)^m u + \lambda G'(u) = F'(u),\\
\noalign{\vskip5pt}
\displaystyle
\int_{\mathbb{R}} K(u) dx = \rho,
\end{cases}
\end{equation*}
in the following cases...
Jacopo Schino, P. Smyrnelis· Topological Methods in Nonli...· 1 citation
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