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Finite linear spaces and projective planes
, 1983
"... In 1948, De Bruijn and Erdös proved that a finite linear space on v points has at least v lines, with equality occurring if and only if the space is either a nearpencil (all points but one collinear) or a projective plane. In this paper, we study finite linear spaces which are not nearpencils. We ..."
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In 1948, De Bruijn and Erdös proved that a finite linear space on v points has at least v lines, with equality occurring if and only if the space is either a nearpencil (all points but one collinear) or a projective plane. In this paper, we study finite linear spaces which are not nearpencils. We obtain a lower bound for the number of lines (as a function of the number of points) for such linear spaces. A finite linear space which meets this bound can be obtained provided a suitable projective plane exists. We then investigate the converse: can a finite linear space meeting the bound be embedded in a projective plane.
A Metric Induced by the Geometric Interpretation of Rolle’s Theorem
"... In this note we discuss a geometric viewpoint on Rolle’s Theorem and we show that a particular setting of the form of Rolle’s Theorem yields a metric that is the hyperbolic metric on the disk. Our result is related to recent developments in the study of Barbilian’s metrization procedure. 1 ..."
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In this note we discuss a geometric viewpoint on Rolle’s Theorem and we show that a particular setting of the form of Rolle’s Theorem yields a metric that is the hyperbolic metric on the disk. Our result is related to recent developments in the study of Barbilian’s metrization procedure. 1