arXiv subjects

Matthew Stephen

Publications and source records attributed to Matthew Stephen.

4 recordsLinked to original sources

arXiv2018

Applications of Gr\"unbaum-type inequalities

Let $1\leq i \leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\subset\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\subset \mathbb{R}^n$: \begin{align*} &V_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} V_i \big( ( K-x) \cap E \big) , \\ &\widetilde{V}_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} \widetilde{V}_i \big( ( K-x) \cap E \big) ; \end{align*} $V_i$ is the $i$th intrinsic volume, and $\widetilde{V}_i$ is the $i$th dual volume taken within $E$. Our results are an extension of an inequality of M. Fradelizi, which corresponds to the case $i=k$. Using the same techniques, we also establish extensions of "Gr\"unbaum's inequality for sections" and "Gr\"unbaum's inequality for projections" to dual volumes.

math.MG↗

arXiv2018

Maximal perimeters of polytope sections and origin-symmetry

Let $P\subset\mathbb{R}^n$ $(n\geq 3)$ be a convex polytope containing the origin in its interior. Let $\mbox{vol}_{n-2} \big( \mbox{relbd} ( P\cap\lbrace t\xi + \xi^\perp \rbrace ) \big)$ denote the $(n-2)$-dimensional volume of the relative boundary of $P\cap\lbrace t\xi + \xi^\perp \rbrace$ for $t\in\mathbb{R}$, $\xi\in S^{n-1}$. We prove the following: if \begin{align*} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\xi^\perp \big) \Big) = \max_{t\in\mathbb{R}} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\lbrace t\xi + \xi^\perp \rbrace \big) \Big) \ \ \forall \ \ \xi\in S^{n-1}, \end{align*} then $P$ is origin-symmetric, i.e. $P = -P$. Our result gives a partial affirmative answer to a conjecture by Makai, Martini, and \'Odor. We also characterize the origin-symmetry of $C^1$ convex bodies in terms of the dual quermassintegrals of their sections; this can be seen as a dual version of the conjecture of Makai et al.

math.MG↗

arXiv2017

Gr\"unbaum's inequality for sections

We show \begin{align*} \frac{ \int_{E \cap \theta^+} f(x) dx }{ \int_E f(x) dx } \geq \left(\frac{k \gamma+1}{(n+1) \gamma+1}\right)^{\frac{k \gamma+1}{\gamma}} \end{align*} for all $k$-dimensional subspaces $E\subset\mathbb{R}^n$, $\theta\in E\cap S^{n-1}$, and all $\gamma$-concave functions $f:\mathbb{R}^n\rightarrow [0,\infty)$ with $\gamma >0$, $0< \int_{\mathbb{R}^n} f(x)\, dx <\infty$, and $\int_{\mathbb{R}^n} x f(x)\, dx$ at the origin $o\in\mathbb{R}^n$. Here, $\theta^+ := \lbrace x\, : \, \langle x,\theta\rangle \geq 0 \rbrace$. As a consequence of this result, we get the following generalization of Gr\"unbaum's inequality: \begin{align*} \frac{ \mbox{vol}_k(K\cap E\cap\theta^+) }{ \mbox{vol}_k(K\cap E) } \geq \left( \frac{k}{n+1} \right)^k \end{align*} for all convex bodies $K\subset\mathbb{R}^n$ with centroid at the origin, $k$-dimensional subspaces $E\subset\mathbb{R}^n$, and $\theta\in E\cap S^{n-1}$. The lower bounds in both of our inequalities are the best possible, and we discuss the equality conditions.

math.MG↗

arXiv2015

Stability results for sections of convex bodies

It is shown by Makai, Martini, and \'Odor that a convex body $K\subset\mathbb{R}^n$, all of whose maximal sections pass through the origin, must be origin-symmetric. We prove a stability version of this result. We also discuss a theorem of Koldobsky and Shane about determination of convex bodies by fractional derivatives of the parallel section function, and establish the corresponding stability result.

math.MG↗