An Introduction To Mensuration And Practical Geometry; With by John Bonnycastle

By John Bonnycastle

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Additional info for An Introduction To Mensuration And Practical Geometry; With Notes, Containing The Reason Of Every Rule

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Auerbach investigates convex sets K which have the property that every chord of some given length, a say, determines an arc of fixed length. He first shows that, if there are such examples, then these same chords determine sectors of fixed area - hence his interest in floating bodies and Ulam's problem. He considers bodies with smooth boundaries and investigates the angle a{9) between the area (and perimeter) bisecting chord in direction 9 and the tangent at the point from which the chord emanates.

In particular, this shows that K i-» F(iir*) is an SL(n) invariant and homogeneous valuation. The next result shows that there are no further examples. 52 Theorem 2 ( 4 6 ). A functional $ : VQ —» R is a measurable, SL(n) invariant valuation which is homogeneous of degree q if and only if there is a constant c € R suc/i t/iat $(P) = < c for q = 0 cV(P) /or q = n cV(P*) for q = —n 0 otherwise 1 /or every P G PQ . It is not known if there are additional examples if $ is not homogeneous. We conjecture that every SL(n) invariant and continuous valuation is a linear combination of a constant, the volume of the body and the volume of the polar body.

Averaging over G, since

Val G Val G , of degree n, such that for

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