Improved bounds for Hadwiger's covering problem via thin-shell estimates

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most $N\left(n\right)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal constant $c>0$. Our approach combines ideas from previous work by Artstein-Avidan and the second named author with tools from Asymptotic Geometric Analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$; in fact, we get the same lower bound for the volume of the intersection of $K$ and $-K$ when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for $N\left(n\right)$ when restricting our attention to convex bodies that are $\psi_{2}$. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity

A note on Santal\'{o} inequality for the polarity transform and its reverse

We prove a Santal\'{o} and a reverse Santal\'{o} inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1 at the origin, also called geometric log-cancave functions. The bounds are sharp up to the optimal universal constants.Comment: 10 pages, 1 figure. Section 5 from the previous version was deleted. A characterization of an equality case in Proposition 2 was adde

Ulam floating bodies

We study a new construction of bodies from a given convex body in Rn which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to p‐affine surface areas. We show that these bodies are related to Ulam’ s long‐standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151836/1/jlms12226_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151836/2/jlms12226.pd

Ulam Floating Body

We study a new construction of bodies from a given convex body in $\mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We show that these bodies are related to Ulam's long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.Comment: 25 pages, 3 figure

Replication, Pathogenesis and Transmission of Pandemic (H1N1) 2009 Virus in Non-Immune Pigs

The declaration of the human influenza A pandemic (H1N1) 2009 (H1N1/09) raised important questions, including origin and host range [1,2]. Two of the three pandemics in the last century resulted in the spread of virus to pigs (H1N1, 1918; H3N2, 1968) with subsequent independent establishment and evolution within swine worldwide [3]. A key public and veterinary health consideration in the context of the evolving pandemic is whether the H1N1/09 virus could become established in pig populations [4]. We performed an infection and transmission study in pigs with A/California/07/09. In combination, clinical, pathological, modified influenza A matrix gene real time RT-PCR and viral genomic analyses have shown that infection results in the induction of clinical signs, viral pathogenesis restricted to the respiratory tract, infection dynamics consistent with endemic strains of influenza A in pigs, virus transmissibility between pigs and virus-host adaptation events. Our results demonstrate that extant H1N1/09 is fully capable of becoming established in global pig populations. We also show the roles of viral receptor specificity in both transmission and tissue tropism. Remarkably, following direct inoculation of pigs with virus quasispecies differing by amino acid substitutions in the haemagglutinin receptor-binding site, only virus with aspartic acid at position 225 (225D) was detected in nasal secretions of contact infected pigs. In contrast, in lower respiratory tract samples from directly inoculated pigs, with clearly demonstrable pulmonary pathology, there was apparent selection of a virus variant with glycine (225G). These findings provide potential clues to the existence and biological significance of viral receptor-binding variants with 225D and 225G during the 1918 pandemic [5]