### A remark on mixed curvature measures for sets with positive reach.

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Negative association for a family of random variables $\left({X}_{i}\right)$ means that for any coordinatewise increasing functions f,g we have $({X}_{i\u2081},...,{X}_{{i}_{k}})g({X}_{j\u2081},...,{X}_{{j}_{l}})\le f({X}_{i\u2081},...,{X}_{{i}_{k}})g({X}_{j\u2081},...,{X}_{{j}_{l}})$ for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem Saxena and Joag-Dev Proschan, and brought to convex geometry in 2005 by Wojtaszczyk Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of...

This paper presents distributional properties of a random cell structure which results from a growth process. It starts at the points of a Poisson point process. The growth is spherical with identical speed for all points; it stops whenever the boundaries of different cells have contact. The whole process finally stops after time t. So the space is not completely filled with cells, and the cells have both planar and spherical boundaries. Expressions are given for contact distribution functions,...

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities ${u}_{0}$ with negative jumps. We show the existence of a stochastic process and a forward flow $\phi $ satisfying ${X}_{s+t}=\phi ({X}_{s},t,{P}_{s},{u}_{s})$ and $\mathrm{d}{X}_{t}=\mathrm{E}[{u}_{0}\left({X}_{0}\right)/{X}_{t}]\mathrm{d}t$, where ${P}_{s}=P{X}_{s}^{-1}$ is the law of ${X}_{s}$ and ${u}_{s}\left(x\right)=\mathrm{E}[{u}_{0}\left({X}_{0}\right)/{X}_{s}=x]$ is the velocity of particle $x$ at time $s\ge 0$. Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map $(x,t)\mapsto M(x,t):=P({X}_{t}\le x)$ is the entropy solution of a scalar conservation law ${\partial}_{t}M+{\partial}_{x}\left(A\left(M\right)\right)=0$ where the flux $A$ represents the particles...

To reconstruct an even Borel measure on the unit sphere from finitely many values of its sine transform a least square estimator is proposed. Applying results by Gardner, Kiderlen and Milanfar we estimate its rate of convergence and prove strong consistency. We close this paper by giving an estimator for the directional distribution of certain three-dimensional stationary Poisson processes of convex cylinders which have applications in material science.

We derive asymptotics for the probability that the origin is an extremal point of a random walk in ${\mathbb{R}}^{n}$. We show that in order for the probability to be roughly $1/2$, the number of steps of the random walk should be between ${\mathrm{e}}^{n/(Clogn)}$ and ${\mathrm{e}}^{Cnlogn}$ for some constant $Cgt;0$. As a result, we attain a bound for the $\frac{\pi}{2}$-covering time of a spherical Brownian motion.