In a series of previous papers ([4], [5], [6], [7]) the Author introduced a studied an intuitive affine extension of the classical notion of a regular n-gon, which, relative to the case of an affine plane over any commutative field, gives a geometric interpretation of the affine regular n-gons introduced with an algebrical procedure by F. Bachmann and E. Schmidt (cfr. [2], p. 162-163). Here we give on outline of these results.
A conjecture stated by D. Veljan (concerning the volume of an n-dimensional simplex) is proved, and an inequality for any set of n + 1 real positive numbers is deduced. (For a general approach to inequalities of the type of the one here obtained, see B.Segre [7]).
It is given by the Introduction.
In an affine plane on GF(q), q = p odd, certain so-called affine regular n-gones are introduced and studied; they exist if; and only if, n divides either q+1 or q-1, or if n=p.
A class of ovals is constructed in any Hall plane of odd order.
The authors give the following characterization of the external lines to an irreducible conic of : If every chord or tangent of an irreducible conic meets a set of points in a unique point, then is necessarily given by all the points of a line external to . While this result admits no analogue in the real field, a number of similar properties can be established or investigated in any Galois geometry.
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