Chiara Bianchini, Andrea Colesanti
We prove a sharp Rogers-Shephard type inequality for the p-difference body of a convex body in the two-dimensional case, for every p greater than or equal to one.
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In this paper we prove a series of Rogers-Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities, or quasi-concave densities attaining their maximum at the origin. Functional versions of classical Rogers-Shephard inequalities are also derived as consequences of our approach.
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of $\ell_p$-diferences of convex bodies whose polar bodies under some condition on the barycenter of their polar bodies.
The difference body operator enjoys different characterization results relying on its basic properties such as continuity, SL(n)-covariance, Minkowski valuation or symmetric image. The Rogers-Shephard and the Brunn-Minkowski inequalities provide upper and lower bounds for the volume of the difference body in terms of the volume of the body itself. In this paper we aim to understand the role of the Rogers-Shephard inequality in characterization results of the difference body and, at the same time, to study the interplay among the different properties. Among others, we prove that the difference …
A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of $n$-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities, with one exception, for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensi…
Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality
研究戈德伯森猜想及\(L_p\)-罗杰斯 - 谢泼德不等式，通过证明凸体与其反射体混合体积由单纯形最大化来证实猜想、细化不等式，还证明凸多面体中单纯形是唯一极值点，并借此证明含原点凸体的\(L_p\)-版本不等式及确定其极值点。
In this paper we study various Rogers-Shephard type inequalities for the lattice point enumerator $\mathrm{G}_{n}(\cdot)$ on $\mathbb{R}^n$. In particular, for any non-empty convex bounded sets $K,L\subset\mathbb{R}^n$, we show that \[\mathrm{G}_{n}(K+L)\mathrm{G}_{n}\bigl(K\cap(-L)\bigr) \leq\binom{2n}{n} \mathrm{G}_{n}\bigl(K+(-1,1)^n\bigr)\mathrm{G}_{n}\bigl(L+(-1,1)^n\bigr). \] and \[ \mathrm{G}_{n-k}(P_{H^\perp} K)\mathrm{G}_{k}(K\cap H)\leq\binom{n}{k}\mathrm{G}_{n}\bigl(K+(-1,1)^n\bigr), \] for $H=\mathrm{lin}\{\mathrm{e}_1,\dots,\mathrm{e}_k\}$, $k\in\{1,\dots,n-1\}$. Additionally, a d…
We provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some new) functional versions of Rogers-Shephard type inequalities as well as some generalizations of the geometric Rogers-Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis-Whitney inequalities. We also obtain a sharp local Loomis-Whitney inequality.
Equality cases for the $L_p$-Rogers--Shephard inequality in the plane and for locally anti-blocking bodies in $\mathbb{R}^n$
研究 $L_p$-Rogers--Shephard 不等式在平面和 $\mathbb{R}^n$ 中局部反阻塞体上的等号情形，证明 $p>1$ 时等号成立当且仅当凸体是以原点为一个顶点的单形。
Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities
该研究建立了Minkowski不对称性的紧稳定性估计，肯定了Belloni和Freund的猜想，并将其应用于改进Banach-Mazur紧集直径的上界。
For a real-valued non-negative and log-concave function we introduce a notion of difference function; the difference function represents a functional analog on the difference body of a convex body. We prove a sharp inequality which bounds the integral of the difference function from above in terms of the integral of the function itself, and we characterize equality conditions. The investigation is extended to an analogous notion of difference function for $α$-concave functions, with $α$ smaller than zero. In this case also, we prove an upper bound for the integral of the $α$-difference functio…
In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets.
In this paper we address the following question: given a measure $μ$ on $\mathbb{R}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H \subset \mathbb{R}^n$ and any convex body $K \subset \mathbb{R}^n$, the following sectional Rogers-Shephard type inequality holds: \[ μ((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} μ(K \cap (H+y))? \] We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m) = \binom{n+m}{m}$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\fr…
L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies
在Firey L_p求和框架下研究经典罗杰斯 - 谢泼德不等式的推广。考虑非对称L_p区域体，证明相关不等式与特定投影体积不等式等价，在p = 2等情况建立不等式及等式条件，还研究了对称凸体情况。
Sharp Rogers--Shephard Inequalities for Rényi Entropy
该研究建立Rényi微分熵的加权Sharp Rogers–Shephard不等式，解决熵Rogers–Shephard猜想，确定等号成立条件及临界阶，通过加权凸交集与指数提升完成证明。
On a generalization of Godbersen's conjecture
本研究提出Godbersen猜想的推广形式，将Schneider对Rogers-Shephard不等式的推广拓展至高阶差体，证明其对反块凸体成立，并建立了与高阶混合体积及Alesker乘积相关的等价表述。
Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies
本文将Schneider的m阶差分体思想推广至投影体、质心体和径向平均体，证明多类仿射等周不等式，并给出有界变差函数的m阶仿射Sobolev不等式。
We obtain a new extension of Rogers-Shephard inequality providing an upper bound for the volume of the sum of two convex bodies $K$ and $L$. We also give lower bounds for the volume of the $k$-th limiting convolution body of two convex bodies $K$ and $L$. Special attention is paid to the $(n-1)$-th limiting convolution body, for which a sharp inequality, which is equality only when $K=-L$ is a simplex, is given. Since the $n$-th limiting convolution body of $K$ and $-K$ is the polar projection body of $K$, these inequalities can be viewed as an extension of Zhang's inequality.
On local Liakopoulos-Meyer type inequalities and their functional counterparts
本文推导了针对凸体和$s$-覆盖的局部Liakopoulos-梅耶型不等式，解决了相关研究中的开放问题。
On the volume of sums of anti-blocking bodies
研究反阻塞体Minkowski和体积的不等式，证明了Plünnecke-Ruzsa型不等式、V.Milman不等式及Roger-Shephard不等式的反演性，给出了精确常数。
In this note we prove some Rogers-Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture.
General Higher Order $L^p$ Mean Zonoids
本文将 $L^p$ 平均带状体从经典Brunn-Minkowski理论推广到高阶Brunn-Minkowski理论框架，并建立了相应的高阶等周不等式。
On unbalanced difference bodies and Godbersen's conjecture
本文探讨了不平衡差体与Godbersen猜想的关系，证明了该猜想的某些结论成立，并展示了Godbersen猜想在平均意义下对任何体都成立，同时推广了Rogers-Shephard不等式。
On Hadwiger's covering problem in small dimensions
本文针对小维度凸体覆盖问题，利用 John 椭球平移、格覆盖及 quermassintegral 估计，给出 $H_5$ 至 $H_8$ 及对称情形 $H_4^s$ 至 $H_6^s$ 的新上界，并讨论改进一般渐近界的方法。
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