Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman, and the Efron-Stein jackknife inequality

ON A QUESTION OF V. I. ARNOL’D

Abstract. We show by a construction that there are at least exp {cV (d−1)/(d+1) } convex lattice polytopes in R d of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. 1. Introduction an

Block partitions: an extended view

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a partition of $S$ into $n$ blocks $B_1, \dots , B_n$ with $|b_i - b_j| \le 1$ for every $i, j$. In this paper, we consider a generalization of the problem in higher dimensions

Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath\'{e}odory theorem and give a Tverberg type theorem for families of convex compacta

How (Not) to Cut Your Cheese

It is well known that a line can intersect at most 2n−1 unit squares of the n × n chessboard. Here we consider the three-dimensional version: how many unit cubes of the 3-dimensional cube [0,n]3 can a hyperplane intersect

Tverberg's Theorem at 50: Extensions and Counterexamples

We describe how a powerful new “constraint method” yields many different extensions of the topological version of Tverberg’s 1966 Theorem in the prime power case— and how the same method also was instrumental in the recent spectacular construction of counterexamples for the general case

Fairly Allocating Contiguous Blocks of Indivisible Items

In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items increases. We also study the efficiency loss of contiguous allocations due to fairness constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game Theory (SAGT), 201

A geoökológia és a geoökológiai térképezés néhány elvi és gyakorlati kérdése

The geographical environment can be investigated from several aspects: - in the biological (ecological) approach emphasis is put on the biotic factors of the environment or on the structure itself; - in the geographical approach research concentrates on the abiotic factors and functions; and - the technological or planning trend focuses the analysis on the economical-technical background of impacts. To distinguish between the first two trends and the related disciplines, the terms (bio) ecology and geoecology are in use. The two concepts differ in handling the role of abiogenic and biogenic factors. In the past decade there was an intension to define geoecology as the study of abiotic factors and of issues concerning the functioning of the physical environment, while landscape ecology investigates the biogenic factors and problems of spatial organisation, structure. Several authors, however, use these concepts interchangeably. The problem is more complicated than that. On the other hand, the concept landscape is narrower or different from that covered by landscape ecology. The latter studies the arrangement of the ecosystem and the flows of matter and energy between its componensts. Here the question is not simply whether or to what extent man-made elements are included in landscape functioning. On the other hand, there is a significant difference between the landscape and the (physical) geographical environment – the true carrier of system properties. This difference of contents was clarified by S. Marosi (1981). In his opinion, the landscape consists of geotopes (naturally including biotopes), while the (geographical) environment is built up of ecotopes and – as a spatial unit – from ecochores. It is the activity of the society related to the socio- or econotopes that makes the geotopes exotopes. In the Marosi model the relationship between landscape and environment is clearly defined. No similar is applied in either the German or in the English-language literature. At the same time, the often used term landscape ecology is difficult to interpret from this standpoint, since they are almost mutually exclusive categories. Spatial pattern is often emphasised in the investigation of the landscape, of the concrete environment and the implications for functioning are neglected, the various ’topes’ are not regarded as aspects of functioning. In the same manner it would be a mistake to restrict the study only to the biogenic or to the abiogenic factors or to disregard functional or system properties. In our opinion – after the scheme by H. Leser (1984) – the German and English schools and the Hungarian views can be reconciled as shown in Fig. l. The size of the landscape ecology frame in the figure may change with various approaches and even it location may vary with the emphasis being on spatiality (like in the Russian literature) or on systems approach (like in the concept of English speaking researchers). Although it contradicts rigid delimitations, geoecology – among others for the above reasons – should cover the analysis of biotic factors too (hence is the uncertainty of delimitation), since they reflect the joint impact of abiotic factors and also point to human influences. Hopefully, the series of examples in the paper call attention to the flexibility of categories. There is communication between them, e.g. geoecology may also reveal structural properties and landscape ecology may answer functional questions of the physical environment. In this respect, the distinction between the two concepts may seem groundless. In our opinion, the in dependent treatment of geoecology separate from landscape ecology, a discipline with more traditions and broader contents, can be justified by the increasing importance of issues of environmental functioning, assessment of the partial potentials of the physical environment (i.e. landscape capacity controlled by landscape budget), data aquisition from field measurements and other practical requirements. The principles of geoecological mapping outlined here (Figure 2) reach beyond the 1:25,000 scale geoecological mapping in Germany, both in methodology and in objective4s. It seemed necessary to apply – in addition to the conventional field surveys, mapping and laboratory techniques – GIS for data storage and processing and for the regional extension of results automated aerial photo interpretation (with scanner) and other remote sensing methods. Although complex systems (such as the landscape) can only be fragmented in a holistic approach, efficiency required the application of a GIS. In the paper three examples are used to illustrate the opportunities to geoecological mapping. The first of them concerns the reclamation or optimal utilisation of surfaces partially used for agricultural purposes, while the second identifies areas affected by hazards, soil erosion, and the third deals with physical loadability through recreation

Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3

In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon in expectation by unilateral deviation. An epsilon well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within epsilon of the best response payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a 2/3 payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least Omega((log n)^(1/3)) in any epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows that there exist epsilon-well-supported equilibria with supports of cardinality O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any delta > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded