M. Fradelizi, M. Madiman, Arnaud Marsiglietti, A. Zvavitch
Let us define for a compact set $A subset mathbb{R}^n$ the sequence $$ A(k) = left{frac{a_1+cdots +a_k}{k}: a_1, ldots, a_kin Aright}=frac{1}{k}Big(underset{k {rm times}}{underbrace{A + cdots + A}}Big). $$ It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that $A(k)$ approaches the convex hull of $A$ in the Hausdorff distance induced by the Euclidean norm as $k$ goes to $infty$. We explore in this survey how exactly $A(k)$ approaches the convex hull of $A$, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on $mathbb{R}^n$, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of $A(k)$ does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets $A$ with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence $A(k)$, and both the Hausdorff distance and effective standard deviation are eventually monotone (once $k$ exceeds $n$). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.
让我们为紧集$asubetmathbb｛R｝^n$定义序列$$a（k）=left｛frac｛a_1+cdots+a_k｝｛k｝：a_1，ldots，a_k in aright｝= frac｛1｝Shapley、Folkman和Starr（1969）以及Emerson和Greenleaf（1969）独立证明，当$k$变为$infty$时，$A（k）$在欧氏范数诱导的Hausdorff距离中接近$A$的凸包。在这项调查中，我们探讨了$A（k）$是如何精确地接近$A$的凸包的，更一般地，探讨了可能不同紧集的Minkowski和是如何接近凸性的，这是通过各种非凸性指数来衡量的。所考虑的非凸性指数包括由$mathbb｛R｝^n$上的任何范数引起的Hausdorff距离、体积亏空（体积的差）、Schneider（1975）引入的非凸指数以及有效标准差或内半径。在首先阐明了这些以前未知或分散在文献中的各种非凸性指数之间的相互关系后，我们发现$A（k）$的体积赤字在维度12或以上不会单调减少到0，从而证伪了Bobkov等人的猜想。（2011），即使他们的猜想在维度1和某些具有特殊结构的集合$A$中被证明是真的。另一方面，Schneider指数在序列$a（k）$上具有强单调性，并且Hausdorff距离和有效标准差最终都是单调的（一旦$k$超过$n$）。在此过程中，我们得到了关于紧集的Minkowski和的体积的新不等式，证伪了Dyn和Farkhi（2004）的一个猜想，证明了我们的结果在组合差异理论中的应用，并提出了一些值得进一步研究的问题。
{"title":"The convexification effect of Minkowski summation","authors":"M. Fradelizi, M. Madiman, Arnaud Marsiglietti, A. Zvavitch","doi":"10.4171/EMSS/26","DOIUrl":"https://doi.org/10.4171/EMSS/26","url":null,"abstract":"Let us define for a compact set $A subset mathbb{R}^n$ the sequence $$ A(k) = left{frac{a_1+cdots +a_k}{k}: a_1, ldots, a_kin Aright}=frac{1}{k}Big(underset{k {rm times}}{underbrace{A + cdots + A}}Big). $$ It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that $A(k)$ approaches the convex hull of $A$ in the Hausdorff distance induced by the Euclidean norm as $k$ goes to $infty$. We explore in this survey how exactly $A(k)$ approaches the convex hull of $A$, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on $mathbb{R}^n$, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of $A(k)$ does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets $A$ with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence $A(k)$, and both the Hausdorff distance and effective standard deviation are eventually monotone (once $k$ exceeds $n$). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EMSS/26","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49230040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
They were for a long time restricted only to the study of natural phenomena or questions pertaining to geometry, before becoming over the course of time, and especially in the last century, a eld in itself. The second half of the XXth century was the golden age" of the exploration of partial dierential equations from a theoretical perspective. The mathematical work of Louis Nirenberg since the early 1950s has to a large extent contributed to the growth of this fundamental area of human knowledge. The name Nirenberg is associated with many of the milestones in the study of PDEs. The award of the Abel Prize to Louis Nirenberg marks a special occasion for us to revisit the development of the eld of PDEs and the work of one of the main actors of its exploration.
{"title":"Exploring the unknown: The work of Louis Nirenberg on partial differential equations","authors":"Tristan Riviere","doi":"10.1090/noti1328","DOIUrl":"https://doi.org/10.1090/noti1328","url":null,"abstract":"They were for a long time restricted only to the study of natural phenomena or questions pertaining to geometry, before becoming over the course of time, and especially in the last century, a eld in itself. The second half of the XXth century was the golden age\" of the exploration of partial dierential equations from a theoretical perspective. The mathematical work of Louis Nirenberg since the early 1950s has to a large extent contributed to the growth of this fundamental area of human knowledge. The name Nirenberg is associated with many of the milestones in the study of PDEs. The award of the Abel Prize to Louis Nirenberg marks a special occasion for us to revisit the development of the eld of PDEs and the work of one of the main actors of its exploration.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2015-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60563949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：