A Hyper-Volume: Problem 84-20Author(s): Richard StanleySource: SIAM Review, Vol. 27, No. 4 (Dec., 1985), pp. 579-580Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2031074 .

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PROBLEMS AND SOLUTIONS 579

B

A

Solution by S. L. PAVERI-FONTANA (Istituto Universitario Navale, Napoli, Italy). Consider a rigid motion on a plane II. At any assigned time t, consider a point

rE IH and its velocity v(r). Then by Chasles' theorem [1], either (i) one has v(r)= w (translation); or (ii) there are a unique IH-perpendicular vector w=# (the angular velocity) and a unique point ce H (the center of instantaneous rotation) such that v(r)= X(r-c) (rotation). Here w, w and c are r-independent but may be t-depen- dent.

Let us examine case (ii). Pick a point r EI H: then r = c iff v(r)= O; when r c, r- c is perpendicular to v, so that the center of instantaneous rotation c must be on the straight line l(r) c H which passes through r and is perpendicular to v(r). Pick r, and r2 E- I 1, with rl, r2 c and rl * r2; then the center of instantaneous rotation must be at the intersection of 1(rl) with 1(r2).

REFERENCE

[1] A. FASANO, V. DE RIENZO AND A. MESSINA, Corso di Meccanica Razionale, Laterza, 1975.

Also solved by J. H. HALTON (University of North Carolina), S. D. HENDRY (Baltimore, MD), M. LEVI (Boston University), J. MADDOCKS (University of Min- nesota), J. A. WILSON (Iowa State University), and the proposer.

Editorial note: Five of the submitted solutions either had invalid assumptions or otherwise were incorrect.

A Hyper-Volume

Problem 84-20, by E. E. DOBERKAT (Clarkson College of Technology). Determine the (n + 1)-dimensional volume of the set of points defined by

T n:s a(XoXl nt Xn)Eh an0alyis : xi-ofxj afor1 _igni.

This problem arose in the analysis of an algorithm.

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580 PROBLEMS AND SOLUTIONS

Comment by RICHARD STANLEY (Massachusetts Institute of Technology). The same problem was proposed by me as problem E2701, American Mathemati-

cal Monthly, 85 (1978), p. 197. A solution by I. G. Macdonald and (independently) R. B. Nelsen was published in vol. 86 (1979), p. 396. The solution may be written in the form (setting Vn +1=volAj)

1 + E VJ'x'=tanx+secx. n21

It is then well known (e.g., L. Comtet, Advanced Combinatorics, pp. 258-9) that n! Vn is equal to the number of alternating permutation a1, a2,---, an of 1,2,--, n, i.e., al <a2> a3<*---

The following is a sketch of another proof of the proposed problem. Define a set

Bn = { (XO,oxl,-** Xn ) E= [0, 1 ] n +1 l ...x x 3_

Define a map 4: An -Bn by 4(x0,x 1,-, x")=(yoYi, y , Yn), where

Y2i:= X2i 9

Y2i+ 1 = maxt X2i+ X2i+1X2i+1 + X2i 2 }

(where if necessary we set xn+1 = 0). Then 4 is easily seen to be a continuous, piecewise linear, volume-preserving bijection. Its inverse is given by

X2i =Y2i,

x2i+1=m1l{Y2i+1?-Y20, Y2i+1-Y2i+2}-

Now there is a canonical way of triangulating Bn into (n + l)-dimensional simplices, such that each simplex corresponds to an alternating permutation ao < a, > a2 < ... of 0, 1, - - , n and has volume 1/(n + 1)!. From this the proof follows.

Postscript: The above proof can be simplified by using Theorem 2.3 of my paper Two poset polytypes, (to appear in Discrete & Computational Geometry) to provide a simpler map An-- Bn, and also this result is generalized.

Also solved by S. L. PAVERI-FONTANA (Istituto Universitario Navale, Napoli, Italy), T. GRANDINE (University of Wisconsin), S. D. HENDRY (Baltimore, MD), B. D. HUGHES (Royal Military College, Canberra, Australia), A. A. JAGERS (Technische Hogeschool Twente, Enschede, the Netherlands), I. I. KOTLARSKI (Oklahoma State University), 0. P. LOSSERS (Eindhoven University of Technology, the Netherlands), T. S. LEWIS (Eastman Kodak Company, Rochester, NY), J. B. LASSERRE (Centre National de la Recherche Scientifique, Toulouse, France), G. MONSTER, (Elmshorn, W. Germany), W. PANNY (Wirtshcaftsuniversitat Wien, Austria), J. A. WILSON (Iowa State University) and the proposer.

Editorial note: J. B. LASSERE, in his solution, first uses a result from his paper, An analytical expression and an algorithm for the volume of a convex polyhedron in Rn, J. Optim. Theory Appl., 39 (1983), to obtain a recurrence relation for An.

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