Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

On the stability of the Motzkin representation of closed convex sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple (C,D) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Irrigation canals in a semi-arid agricultural landscape surrounded by wetlands: Their role as a habitat for birds during the breeding season

Traditional water supply systems in semi-arid agrarian ecosystems, mainly irrigation canals, contribute to the diversity of the landscape and influence the composition of species. To evaluate their effect on bird communities in the breeding season, we selected a rural area in southeastern Spain, where an intricate and extensive network of irrigation canals and cultivated areas is located between two wetlands declared as Natural Parks. Birds were counted at representative points distributed throughout the canal network at which we recorded several variables related to the physical features, the vertical and horizontal structure of associated vegetation, reed development (Phragmites australis) and land use in the neighboring areas. We detected 37 bird species, most of which were also breeding in the wetlands nearby. We used Hierarchical Partitioning analyses to identify the variables most strongly related to the probability of the presence of selected species and species richness. Vegetation cover and height close to the canals, together with reed development, were the most important types of variables explaining species presence and richness. We found that current management practices for reeds in canals are not well-suited for biodiversity conservation. We therefore propose alternatives that could be implemented in the area in cooperation with stakeholders.