Some functional forms of Blaschke-Santal\'o inequality

We establish new functional versions of the Blaschke-Santal\'o inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K. Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case.Comment: 19 pages, to appear in Mathematische Zeitschrif

On the volume of sections of a convex body by cones

Let $K$ be a convex body in $\mathbb R^n$. We prove that in small codimensions, the sections of a convex body through the centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies.Comment: 13 page

Inventarisatie van archeologische resten uit de Eerste Wereldoorlog en de mogelijkheden voor onderzoek naar sporen van andere conflicten: Battlefield Archaeology in West-Vlaanderen

The province of Western Flanders contains a particular kind of heritage: the World War I remains. The German-Allied frontline (the Western Front), which crossed Belgium from Nieuwpoort (Nieuport) to Mesen (Messines), is world famous. The Ypres Salient is the best known part of it. Less known but at least as well preserved are the coastal defences and the Hollandstellung, which is a defensive line along the Belgian-Dutch border. Making an inventory of this heritage (consisting of the remains of missing soldiers, trenches, metre gauge railways, (deep) dugouts, concrete shelters, etc.) is not an easy task. Therefore the study of the so-called trenchmaps (maps which were made during the war and indicating trenches, metre-gauge railways, wire entanglement, etc.) and aerial photographs from the Great War are important sources of information. Parts of them have been indicated in GIS (Geographic Information System) and the CAI (Central Archaeological Inventary). In 2001 a first attempt was made to make an overview of the archaeological World War One heritage in a couple of villages: Houthulst, Klerken, Jonkershove and Merkem. This region is not a part of the Ypres Salient, but the German frontline follows the border from Merkem along the Yser River and the Ieperlee Canal (this stream runs from the Yser to Ypres). Klerken, Jonkershove and Houthulst were part of the German backlines. The forest of Houthulst was used as the headquarters of the Germans for both the Yser Front and the Ypres Salient. It is clear that the region was an important area during the Great War. At first an inventory was made of the classic archaeological heritage (e.g. Prehistoric, Roman, Medieval remains). In a next phase trenchmaps and aerial photographs were used to locate the World War One remains. Locating the huge amount of structures on modern maps proved to be a very time-consuming but very interesting task. Besides that, trenchmaps did not seem to be accurate enough to make a decent inventory of them. At the time there wasnt enough time or money to make a full inventory of World War One remains in Flanders. That is why the CAI-team focussed on the seriously threatened areas. Several techniques had to be tried out. The first opportunity came as a result of the A19 project, the first big scale battlefield archaeology project in Flanders. By order of the former minister of Interior Affairs, Culture, Youth, and Civil Administration for the Flemish Community, Paul Van Grembergen, the VIOE (Vlaams Instituut voor het Onroerend Erfgoed Flemish Heritage Institute) was put in charge of evaluating the archaeological heritage on the (possible) future A19 motorway trajectory. The study of the World War One heritage was going to be especially important, because the threatened area consists of an important part of the Ypres Salient. Here the combination of trenchmaps, fieldwalking and aerial photography was crucial for the assessment of the archaeological value of the area and for planning and achieving excavations of nine different sites. These excavations started in November 2002. The combination of an intensive GIS study with fieldwork proved to be useful. The first step was to locate all the major battle lines and hinterland of the different armies involved. Therefore some trenchmaps and aerial photographs were used, combined with several written sources. With this information a large campaign of fieldwalking was started on the future A19-track. Farmers and (ex-) inhabitants of the region were interviewed about their knowledge of World War One remains in the area. This also gave us some very valuable information. Everything has been drawn in several GIS themes. Based on that and based on a study written by Prof. P. Chasseaud from the University of Greenwich (Department Of Military Cartography) nine zones were selected on which further research had to be done. Until now, some of those areas have been excavated. Other areas have been studied using resistivity and magnetometry surveys. The aerial photographs used for the study are British aerial reconnaissance pictures, dating from the First World Warand are from the Imperial War Museums Box Collection in London and the In Flanders Fields Museum in Ypres. Most of those pictures have been georeferenced on modern topographic maps (scale 1:10,000). Trenches, barbed wire, tracks, old field boundaries, battery positions and so on could be located exactly on the map. This is of course a very rich source of information, not only to select zones to excavate, but also to help to interpret the trenches and other structures which are excavated or located during the geophysical survey. The large amounts of pictures which are available for a small area show us the history and evolution of the war in its geographical (and archaeological!) context. Of course, it is also an indication of the complexity of the research, given the continuing change of structures. The results of the archaeological fieldwork are very promising: well-preserved trenches and duckboards, ammunition depots, all kinds of artifacts, human remains, etc. All this shows that the use of trenchmaps and aerial photographs, combined with GIS, can provide us a lot of information. They should be used in all archaeological research about the Great War, or Battlefield Archaeology in general, whenever possible. To make the research about World War One remains more organized, a new battlefield Archaeology- unit was set up in November 2003, a co-operation between the VIOE and several partners. In 2004 this department was transformed into the Association for World War Archaeology (A.W.A.). Also in 2004, a large scale inventory project was launched. The Ypres Salient and other frontlines are inventorized using aerial photographs and trenchmaps in a GIS system. This new project is a co-operation between the In Flanders Fields Museum (IFFM Ypres) and the Flemish Heritage Institute (VIOE)