Estimates for measures of sections of convex bodies

A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these inequalities from stability in comparison problems for different generalizations of intersection bodies

Refractive change following pseudophakic vitrectomy: a retrospective review

Background To assess the occurrence and magnitude of refractive change in pseudophakic eyes undergoing 20 gauge pars plana vitrectomy without scleral buckling and to investigate possible aetiological factors. Methods Retrospective case note review of 87 pseudophakic eyes undergoing 20 gauge pars plana vitrectomy for a variety of vitreo-retinal conditions over a three-year period. Anterior chamber depth (ACD) was measured before and after vitrectomy surgery in 32 eyes. Forty-three pseudophakic fellow eyes were used as controls. Results Eighty-seven eyes (84 patients) were included in the study. Mean spherical equivalent refraction prior to vitrectomy was -0.20 dioptres, which changed to a mean of -0.65 dioptres postoperatively (standard deviation of refractive change 0.59, range-2.13 to 0.75 dioptres) (p < 0.001). Sixty-one of the 87(70%) eyes experienced a myopic shift and 45(52%) eyes had a myopic shift of -0.5 dioptres or more. Mean fellow eye refraction was -0.19 dioptres preoperatively and -0.17 dioptres postoperatively (p = 0.14)(n = 37) Mean ACD preoperatively was 3.29 mm and postoperatively 3.27 mm (p = 0.53) (n = 32) and there was no significant change in ACD with tamponade use. Regression analysis revealed no statistically significant association between changes in anterior chamber depth, as well as a wide variety of other pre-, intra and postoperative factors examined, and the refractive change observed. Conclusion Significant refractive changes occur in some pseudophakic patients undergoing 20 g pars plana vitrectomy. The mean change observed was a small myopic shift but the range was large. The aetiology of the refractive change is uncertain

Mid-infrared passively switched pulsed dual wavelength Ho3+ -doped fluoride fiber laser at 3 μm and 2 μm

Cascade transitions of rare earth ions involved in infrared host fiber provide the potential to generate dual or multiple wavelength lasing at mid-infrared region. In addition, the fast development of saturable absorber (SA) towards the long wavelengths motivates the realization of passively switched mid-infrared pulsed lasers. In this work, by combing the above two techniques, a new phenomenon of passively Q-switched ~3 μm and gain-switched ~2 μm pulses in a shared cavity was demonstrated with a Ho3+-doped fluoride fiber and a specifically designed semiconductor saturable absorber (SESAM) as the SA. The repetition rate of ~2 μm pulses can be tuned between half and same as that of ~3 μm pulses by changing the pump power. The proposed method here will add new capabilities and more flexibility for generating mid-infrared multiple wavelength pulses simultaneously that has important potential applications for laser surgery, material processing, laser radar, and free-space communications, and other areas

Quantitative Helly-Type Theorem for the Diameter of Convex Sets

We provide a new quantitative version of Helly’s theorem: there exists an absolute constant α&amp;gt; 1 with the following property. If { Pi: i∈ I} is a finite family of convex bodies in Rn with int (⋂ i ∈ IPi) ≠ ∅ , then there exist z∈ Rn, s⩽ αn and i1, … is∈ I such that (Formula Presented.) where c&amp;gt; 0 is an absolute constant. This directly gives a version of the “quantitative” diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound O(n3 / 2) can be improved to O(n). © 2016, Springer Science+Business Media New York

Brascamp-lieb inequality and quantitative versions of helly&apos;s theorem

We provide new quantitative versions of Helly&apos;s theorem. For example, we show that for every family of closed half-spaces in such that has positive volume, there exist and such that where 0$]]&gt; are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John&apos;s decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp-Lieb inequality and an appropriate variant of Ball&apos;s proof of the reverse isoperimetric inequality. © 2016 University College London

Continuous Version of the Approximate Geometric Brascamp–Lieb Inequalities

Given γ&amp;gt; 1 we say that a Borel measure ν on Sn-1 is a γ-approximation of an isotropic measure if In⪯Tν=∫Sn-1u⊗udν(u)⪯γIn,where In is the identity matrix. We provide a generalization of Barthe’s continuous version of the Brascamp–Lieb inequalities to the context of these approximate isotropic measures, and we apply these inequalities to obtain stability results for some classical positions of convex bodies. © 2022, Mathematica Josephina, Inc

Vector-valued Maclaurin inequalities

We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type inequality for parallelepipeds. © 2021 World Scientific Publishing Company

Sub-Gaussian directions of isotropic convex bodies

Let K be a centered convex body of volume 1 in Rn. A direction θ∈Sn-1 is called sub-Gaussian for K with constant b&amp;gt;0 if {norm of matrix}〈{dot operator},θ〉{norm of matrix}Lψ2(K)≤b{norm of matrix}〈{dot operator},θ〉{norm of matrix}2. We show that if K is isotropic then most directions are sub-Gaussian with a constant which is logarithmic in the dimension. More precisely, for any a&amp;gt;1 we have, for all θ in a subset Θa of Sn-1 with σ(Θa)≥1-n-a, where C&amp;gt;0 is an absolute constant. © 2015 Elsevier Inc