we determine the infimum of the areas of the simple n-gons in the Euclidean plane, having sides of length a1,...,an (in some order). The infimum is attained (in limit) if the polygon degenerates into a certain kind of triangle, plus some
parts of zero area. We show the same result for simple polygons on the sphere (of not too great length), and for simple polygons
in the hyperbolic plane. Replacing simple n-gons by convex ones, we answer the analogous questions. The infimum is attained also here for degeneration into a certain
kind of triangle.
We prove the theorem mentioned in the title for ℝn where n ≧ 3. The case of the simplex was known previously. Also the case n = 2 was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic n-spaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas and some partial results about sufficient conditions for the existence of (convex) tetrahedra.
Hungarian fruit vinegars were characterised in terms of physicochemical attributes (total polyphenol content, antioxidant characteristics/FRAP, CUPRAC, ABTS/, ascorbic acid content, pH, total soluble solids), sensory profiles, and antimicrobial properties.
Both compositional and sensory profiles showed distinct patterns depending on the type of vinegar (Tokaj wine, balsamic or apple) and the additional fruit used. Balsamic vinegars maturated on rosehip, sea buckthorn, and raspberry showed outstanding antioxidant performances. Rosehip, raspberry, and quince vinegars, as well as vinegars produced from Tokaji aszú and balsamic apple obtained high scores for fruity and sweet notes.
Antimicrobial activities were tested on Gram-negative and Gram-positive organisms, including probiotic bacteria. Generally, only weak activities were obtained, which was attributed to the natural sugar content of the samples, depending on the type of the vinegar and the fruit. Similar results, but more pronounced bacterial growth inhibitions were obtained for probiotic strains, however, some probiotic strains were resistant to at least two of the vinegars. Based on these, balsamic apple, raspberry, rosehip, quince, and sea buckthorn may qualify as potential functional components of probiotic preparations containing some of the strains tested.