Let D be a set of n pairwise disjoint unit balls in R and P the set of their center points. A hyperplane H is an m-separator for D if each closed halfspace bounded by H contains at leastm points from P. This generalizes the notion of halving hyperplanes, which correspond to n/2-separators. The analogous notion for point sets has been well studied. Separators have various applications, for insta...

We show that a separable proximinal subspace of X, say Y is strongly proximinal (strongly ball proximinal) if and only if Lp(I, Y ) is strongly proximinal (strongly ball proximinal) in Lp(I,X), for 1 ≤ p <∞. The p =∞ case requires a stronger assumption, that of ’uniform proximinality’. Further, we show that a separable subspace Y is ball proximinal in X if and only if Lp(I, Y ) is ball proximin...

Let B be a set of n unit balls in R3. We show that the combinatorial complexity of the space of lines in R3 that avoid all the balls of B is O(n3+ε), for any ε > 0. This result has connections to problems in visibility, ray shooting, motion planning, and geometric optimization.

This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors is changed according to the number of black balls among them.The limiting behaviour of the composition of the urn when both the tim...

We construct a class of envelope surfaces in R, more precisely envelopes of balls. An envelopesurface is a closed C (tangent continuous) manifold wrapping tightly around the union of aset of balls. Such a manifold is useful in modeling since the union of a finite set of balls canapproximate any closed smooth manifold arbitrarily close.The theory of envelope surfaces generali...

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce new class orientable manifolds call filling enlargeable and prove: If $M$ all unit balls in universal cover $(V,g)$ have volume less than constant $\frac{1}{2}\varepsilon_n$, then $width(V,g)\le...