![]() ![]() That is, if two points ( and ) have masses and, respectively, a third point between and which divides into the ratio will have mass. If two points are balanced, the point on the balancing line used to balance them has a mass of the sum of the masses of the two points. If two points balance, the product of the mass and distance from a line of balance of one point will equal the product of the mass and distance from the same line of balance of the other point. Any line passing this central point will balance the figure. From the first weight, others can be derived using a few simple rules. From there, WLOG a first weight can be assigned. The way to systematically assign weights to the points involves first choosing a point for the entire figure to balance around. Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever). Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths must be inversely proportional to their weight (just like a balanced lever). The technique greatly simplifies certain problems. The technique did not catch on until the 1960s when New York high school students made it popular. ![]() Mass point geometry was invented by Franz Mobius in 1827 along with his theory of homogeneous coordinates. Mass points are generalized by barycentric coordinates. In essence, it involves using a local coordinate system to identify points by the ratios into which they divide line segments. Our main result there is a condition on the modulus of asymptotic smoothness \(\overline(t) > 0\) for all \(t > 0.\) Clearly, a dual space which is AUC \(^*\) is also AUC.Mass points is a technique in Euclidean geometry that can greatly simplify the proofs of many theorems concerning polygons, and is helpful in solving complex geometry problems involving lengths. We end the section with the necessary background on asymptotic uniform properties of Banach spaces. As a warm up, we give a simple proof that uniformly non-square spaces do not admit \(\Delta\)-points and explain why simple considerations on the diameter of slices cannot rule out \(\Delta\)-points outside of this setting. 2 we recall the notion of slices, give the definition of Daugavet- and \(\Delta\)-points, and state a few simple geometric lemmata. Let us now describe the content of the paper and expose our main results. ![]() We provide new examples of Banach spaces failing to contain \(\Delta\)-points and we introduce weaker notions which can be viewed as a step forward in the direction of constructing an example of a superreflexive space with a Daugavet- or a \(\Delta\)-point. In the present paper, we continue the investigation of Daugavet- and \(\Delta\)-points in general Banach spaces by focusing on the interactions between those points and the asymptotic geometry of the space. Extending this result to the general setting, Veeorg was then able to provide in a surprising example of a metric space whose free space has the Radon–Nikodým property (RNP) and admits a Daugavet-point. It was observed that Daugavet-points have to be at distance 2 from every denting point of the unit ball in any given Banach space, and it was proved that the converse holds in every free-space in the compact setting. The study of Daugavet- and \(\Delta\)-points in this context started in where a characterization of Daugavet-points in free-spaces over compact metric spaces was discovered. Īnother striking example illustrating this was obtained in the context of Lipschitz-free spaces. For example, there exists a Banach space with a 1-unconditional basis such that the set of Daugavet-points are weakly dense in the unit ball. It soon appeared that even nice properties which on a global level prevent the space to have the Daugavet property or the DLD2P do not provide an obstruction to the existence of Daugavet- or \(\Delta\)-points in the space. In particular, a strong emphasis was put on finding linear or geometric properties that would prevent norm one elements in a space to be Daugavet- or \(\Delta\)-points. 2 for precise definitions and equivalent reformulations.įrom their introduction on, Daugavet- and \(\Delta\)-points attracted a lot of attention and were intensively studied in classical Banach spaces. Daugavet- and \(\Delta\)-points first appeared in as natural pointwise versions of geometric characterizations of the Daugavet property and of the so-called spaces with bad projections (also known as spaces with the diametral local diameter two property (DLD2P) ).
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