We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.

In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.

We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n). This improves an earlier bound of O(1.6181n) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

Based on the definition of divisibility of Markovian quantum dynamics, we discuss the Markovianity of tensor products, multiplications and some convex combinations of Markovian quantum dynamics. We prove that the tensor product of two Markovian dynamics is also a Markovian dynamics and propose a new witness of non-Markovianity.

We propose a new two-phase method for reconstruction of blurred images corrupted by impulse noise. In the first phase, we use a noise detector to identify the pixels that are contaminated by noise, and then, in the second phase, we reconstruct the noisy pixels by solving an equality constrained total variation minimization problem that preserves the exact values of the noise-free pixels. For images that are only corrupted by impulse noise (i.e., not blurred) we apply the semismooth Newton's method to a reduced problem, and if the images are also blurred, we solve the equality constrained reconstruction problem using a first-order primal-dual algorithm. The proposed model improves the computational efficiency (in the denoising case) and has the advantage of being regularization parameter-free. Our numerical results suggest that the method is competitive in terms of its restoration capabilities with respect to the other two-phase methods.

Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc.121(4) (1994), 1063–1073], we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^{d}d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^{d}\binom{2d}{d}d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that, in the latter case, one can always find families of at least $\unicode[STIX]{x1D6FA}((2/\sqrt{3})^{d})$ translates of $K$ with the above property.

Let $X$ be a vector space and let $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$ be an extended real-valued function. For every function $f:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$, let us define the $\unicode[STIX]{x1D711}$-envelope of $f$ by

$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}(x)=\sup _{y\in X}\unicode[STIX]{x1D711}(x-y)\begin{array}{@{}c@{}}-\\ \cdot \end{array}f(y),\end{eqnarray}$$

$\begin{array}{@{}c@{}}-\\ \\ \\ \cdot \end{array}$

$\mathbb{R}\cup \{-\infty ,+\infty \}$

$f\mapsto f^{\unicode[STIX]{x1D711}}$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$

$\unicode[STIX]{x1D706}>0$

$p\geqslant 1$

$\unicode[STIX]{x1D706}$

$p$

$\unicode[STIX]{x1D711}$

$-\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D711}$

$X$

$\unicode[STIX]{x1D6EC}$

$C\subset X$

$\unicode[STIX]{x1D6EC}$

$C$

$C^{\unicode[STIX]{x1D6EC}}=\bigcap _{x\in C}(x+\unicode[STIX]{x1D6EC})$

$C\mapsto C^{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x1D711}$

Let $n\geqslant C$ for a large universal constant $C>0$ and let $B$ be a convex body in $\mathbb{R}^{n}$ such that for any $(x_{1},x_{2},\ldots ,x_{n})\in B$, any choice of signs $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2},\ldots ,\unicode[STIX]{x1D700}_{n}\in \{-1,1\}$ and for any permutation $\unicode[STIX]{x1D70E}$ on $n$ elements, we have $(\unicode[STIX]{x1D700}_{1}x_{\unicode[STIX]{x1D70E}(1)},\unicode[STIX]{x1D700}_{2}x_{\unicode[STIX]{x1D70E}(2)},\ldots ,\unicode[STIX]{x1D700}_{n}x_{\unicode[STIX]{x1D70E}(n)})\in B$. We show that if $B$ is not a cube, then $B$ can be illuminated by strictly less than $2^{n}$ sources of light. This confirms the Hadwiger–Gohberg–Markus illumination conjecture for unit balls of $1$-symmetric norms in $\mathbb{R}^{n}$ for all sufficiently large $n$.

We provide new quantitative versions of Helly’s theorem. For example, we show that for every family $\{P_{i}:i\in I\}$ of closed half-spaces in $\mathbb{R}^{n}$ such that $P=\bigcap _{i\in I}P_{i}$ has positive volume, there exist $s\leqslant \unicode[STIX]{x1D6FC}n$ and $i_{1},\ldots ,i_{s}\in I$ such that

$$\begin{eqnarray}\operatorname{vol}_{n}(P_{i_{1}}\cap \cdots \cap P_{i_{s}})\leqslant (Cn)^{n}\operatorname{vol}_{n}(P),\end{eqnarray}$$

$\unicode[STIX]{x1D6FC},C>0$

This paper deals with long-range dependence of random measures on ℝd. By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived.

The space of shapes of quadrilaterals can be identified with $\mathbb{CP}^{2}$. We deal with the subset of $\mathbb{CP}^{2}$ corresponding to convex quadrilaterals and the subset which corresponds to simple (that is, without self-intersections) quadrilaterals. We provide a complete description of the topological closures in $\mathbb{CP}^{2}$ of both spaces. Although the interior of each space is homeomorphic to a disjoint union $\mathbb{R}^{4}\sqcup \mathbb{R}^{4}$, their closures are topologically different. In particular, the boundary of the space corresponding to convex quadrilaterals is homeomorphic to a pair of three-dimensional spheres glued along a Möbius strip while the boundary of the space corresponding to simple quadrilaterals is more complicated.

This paper introduces a two-stage model for multi-channel image segmentation, which is motivated by minimal surface theory. Indeed, in the first stage, we acquire a smooth solution u from a convex variational model related to minimal surface property and different data fidelity terms are considered. This minimization problem is solved efficiently by the classical primal-dual approach. In the second stage, we adopt thresholding to segment the smoothed image u. Here, instead of using K-means to determine the thresholds, we propose a more stable hill-climbing procedure to locate the peaks on the 3D histogram of u as thresholds, in the meantime, this algorithm can also detect the number of segments. Finally, numerical results demonstrate that the proposed method is very robust against noise and superior to other image segmentation approaches.

We generalize to the anisotropic case some classical and recent results on the (n – 1)-Minkowski content of rectifiable sets in ℝn, and on the outer Minkowski content of subsets of ℝn. In particular, a general formula for the anisotropic outer Minkowski content is provided; it applies to a wide class of sets that are stable under finite unions.

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

In this paper we give a proof of the Green–Osher inequality in relative geometry using the minimal convex annulus, including the necessary and sufficient condition for the case of equality.

Let K ⊂ ℝN be any convex body containing the origin. A measurable set G ⊂ ℝN with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r > 0, the measure of G ⋂ (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). In a previous work, we proved for the case 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 ℝN: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, which builds upon results obtained in our previous work, relies on an asymptotic formula for the measure of G ⋂ (x + rK) for large values of the parameter r and a classical characterization of ellipsoids due to Petty.

In a $d$ -dimensional convex body $K$ random points $X_{0},\ldots ,X_{d}$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion if $K\subset L$ implies that the expected volume of a random simplex in $K$ is smaller than the expected volume of a random simplex in $L$ . Continuing work of Rademacher [On the monotonicity of the expected volume of a random simplex. Mathematika58 (2012), 77–91], it is shown that moments of the volume of random simplices are, in general, not monotone under set inclusion.

We prove that the log-Brunn–Minkowski inequality (log-BMI) for the Lebesgue measure in dimension $n$ would imply the log-BMI and, therefore, theB-conjecture for any even log-concave measure in dimension $n$ . As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure in the plane. Moreover, we prove that the log-BMI reduces to the following: for each dimension $n$ , there is a density $f_{n}$ , which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_{n}$ . As a byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_{p}\cdot ~\text{e}^{t}L)^{\circ }|$ , where $p\geqslant 1$ and $K$ , $L$ are symmetric convex bodies, which we are able to prove in some instances and, as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

One of the most fruitful results from Minkowski’s geometric viewpoint on number theory is his so-called first fundamental theorem. It provides an optimal upper bound for the volume of a $0$ -symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of a $0$ -symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.

The illumination problem may be phrased as the problem of covering a convex body in Euclidean $n$-space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball, there is a centrally symmetric convex body of illumination number exponentially large in the dimension.

We investigate the problem of the decomposition of balls into a finite number of congruent pieces in dimension $d=2k$. In addition, we prove that the $d$-dimensional unit ball $B_{d}$ can be divided into a finite number of congruent pieces if $d=4$ or $d\geqslant 6$. We show that the minimal number of required pieces is less than $20d$, if $d\geqslant 10$.