More on logarithmic sums of convex bodies

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, 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 byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_p\cdot e^tL)^{\circ}|$, where $p\geq 1$ 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.Comment: Minor corrections, some additional references, agnowledgemen

Wulff shapes and a characterization of simplices via a Bezout type inequality

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K).$$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset {\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies

Characterization of Simplices via the Bezout Inequality for Mixed volumes

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ and any convex bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. In this paper we prove that this is indeed the case if we assume that $\Delta$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots, K_r$ then the boundary of $\Delta$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.Comment: 8 page

European Ageing Population and Asylum Seekers: Can Europe Solve One Problem with Another?

One of the biggest challenges of our times is the ageing of the population. In the Western world - especially in Europe - low fertility rates and higher life expectancy have shrunk the labor force and created a large pool of older people who have to depend on that decreasing labor force. The economic ramifications of the situation can be severe. When the population is ageing, pension and healthcare costs are raising. Meanwhile, the labor force - the group that contains the people who are paying for those costs - is shrinking. As a result of the reduced labor force, the labor supply decreases. Like falling dominoes, different aspects of the economy underperform, eventually leading to reduced growth

Rockfall hazard in Greece.

Η γεωλογική δομή της Ελλάδας (συχνή εμφάνιση βραχωδών σχηματισμών, παρουσία ρηγμάτων και κερματισμός των πετρωμάτων) σε συνδυασμό με το απότομο και ορεινό ανάγλυφο καθώς και την υψηλή σεισμικότητα, συμβάλλουν στην υψηλή διακινδύνευση έναντι καταπτώσεων βράχων. Τις τελευταίες δεκαετίες, οι καταπτώσεις βράχων είναι συχνό φαινόμενο στον Ελλαδικό χώρο εξαιτίας της αύξησης των ακραίων βροχοπτώσεων καθώς και τις επέκτασης της ανθρώπινης δραστηριότητας. Το άρθρο παρουσιάζει την διακινδύνευση έναντι καταπτώσεων στην Ελλάδα με τη χρήση μιας βάσης δεδομένων και προσδιορίζει τη σχέση συγκεκριμένων παραμέτρων, όπως: α) μηχανισμός γένεσης (βροχόπτωση, σεισμός), β) κλίση πλαγιάς, γ) λιθολογία, δ) παρουσία ρήγματος, ε) μέγεθος πίπτωντων τεμαχών με την πιθανότητα εκδήλωσης αυτών με χρήση στατιστικής προσέγγισης. Διερευνάται επίσης η χρονική και χωρική συχνότητα και τέλος η επίπτωση των καταπτώσεων στις ανθρώπινες δραστηριότητες (δρόμους, κατοικημένες περιοχές, αρχαιολογικοί χώροι)The geological structure of Greece (frequent occurrence of rock formations, existence of faults and fracturing of rocks), the steep topography and mountainous terrain as well as its high seismicity, creates a significant rockfall hazard. During the last decades, rockfalls in Greece are becoming a frequent phenomenon due to the increase of intense rainfall events but also due to the extension of human activities in mountainous areas.   The paper presents rockfall hazard in Greece trough an inventory of rockfalls and investigates  the  correlation  of  specific  factors,  namely:  a)  triggering  mechanism (rainfall, seismicity), b) slope angle, c) lithology, d) fault presence, e) block size in the  probability of occurrence of these, based on a statistical approach. The time and space frequency of the events is also investigated. Finally, the impact of the events on human and infrastructures (transportation infrastructure, inhabited areas, archaeological sites) is discussed