Simple estimates for ellipsoid measures

Let B denote a three-dimensional body of rotation, with respect to one coordinate axis, whose boundary is sufficiently smooth and of bounded nonzero Gaussian curvature throughout, except for the two boundary points on the axis of rotation, where the curvature may vanish. For a large real variable t, we are interested in the number A(t) of integer points in the linearly dilated body tB, in particular in the lattice discrepancy P(t) = A(t) - volume(tB). We are able to evaluate ...

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn-Minkowski in...

We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume $\varepsilon$ ($0 \leq \varepsilon \leq 1/2$) inside a $n$-dimensional ball $U$ of unit volume that implements among all the sets of volume $\varepsilon$ is a set with the smallest possible free surface area, lying in one half-space with respect to a...

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the...

Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \varepsilon}$, for simplexes and hyperoctahedrons $-$ of order $-\ln \varepsilon$, for $\ell_p$ balls with $p \in [1;2]$ $-$ of order $(-\ln \varepsilon)^{\frac{1}{p}}$. These estimates ...

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we also address Topping's conjecture relating diameter and mean curvature for connect...

We estimate the linear isoperimetric constants of an n-dimensional ellipse. Using these estimates and a technique of Gromov, we estimate the Hopf and linking invariants of Lipschitz maps from ellipses to round spheres. Using these estimates, we give a lower bound for the k-dilation of degree non-zero maps between ellipses.

In the present paper we establish area and volume estimates for spacetimes satisfying the strong energy condition in terms of the area and the $L^n$-norm of the second fundamental form or the mean curvature of an initial Cauchy hypersurface. We believe that these estimates will lay some of the groundwork in establishing new convergence results for Cauchy developments $(M_j, g_j)$ of suitably converging initial data $(\Sigma_j ,h_j ,K_j )$.

Let K\subset R^N be any convex body containing the origin. A measurable set G\subset R^N with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r>0, the measure of G\cap (x+r K) is constant when x varies on the boundary of G (here, x+r K denotes a translation of a dilation of K). In [6], we proved for the case in which N=2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in...

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to \Lambda. We will prove here that the number of lattice points inside K can also accept an upper bound of roughly the same size, in the special case where K is an ellipsoid. Whether this is also true for all K unconditionally is an open problem, b...