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Register a new userStrengthened inequalities for the mean width and the $ell$-norm
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Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $ell$-norm of convex bodies whose Lowner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschlager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Lowner ellipsoid is the Euclidean unit ball. Here we prove stronger stabili
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Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of ``essentiality, our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.
In this paper, we study several weighted norm inequalities for the Opdam--Cherednik transform. We establish differe
We develop a methodology for the computation of the $Kto ell u_ell ell^+ ell^-$ decay width using lattice QCD and present an exploratory study here. We use a scalar function method to account for the momentum dependence of the decay amplitude and adopt the infinite volume reconstruction (IVR) method to reduce the systematic errors such as the temporal truncation effects and the finite-volume effects. We then perform a four-body phase-space integral to obtain the decay width. The only remaining technical problem is the possible power-law finite-volume effects associated with the process of $Ktopipi ell u_ellto ell u_ell ell^+ ell^-$, where the intermediate state involves multiple hadrons. In this work, we use a gauge ensemble of twisted mass fermion with a pion mass $m_pi=352$ MeV and a nearly-physical kaon mass. At this kinematics, the $pipi$ in the intermediate state cannot be on shell simultaneously as $2m_pi>m_K$ and the finite-volume effects associate with $pipi$ state are exponentially suppressed. Using the developed methods mentioned above, we calculate the branching ratios for four channels of $Kto ell u_ellell^+ ell^-$, and obtain the results close to the experimental measurements and ChPT predictions. Our work demonstrates the capability of lattice QCD to improve Standard Model prediction in $Kto ell u_ell ell^+ ell^-$ decay width.
If a convex body $K subset mathbb{R}^n$ is covered by the union of convex bodies $C_1, ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two parallel hyperplanes that sandwich $K$) and the inradius (the largest radius of a ball contained in $K$): the sum of the widths of the $C_i$ is at least the width of $K$ (this is the plank theorem of Thoger Bang), and the sum of the inradii of the $C_i$ is at least the inradius of $K$ (this is due to Vladimir Kadets). We adapt the existing proofs of these results to prove a theorem on coverings by certain generalized non-convex multi-planks. One corollary of this approach is a family of inequalities interpolating between Bangs theorem and Kadetss theorem. Other corollaries include results reminiscent of the Davenport--Alexander problem, such as the following: if an $m$-slice pizza cutter (that is, the union of $m$ equiangular rays in the plane with the same endpoint) in applied $N$ times to the unit disk, then there will be a piece of the partition of inradius at least $frac{sin pi/m}{N + sin pi/m}$.
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