We show that the volume of the projection body $\Pi(Z)$ of an $n$-dimensional zonotope $Z$ with $n+1$ generators and of volume $1$ is always exactly $2^n$. Moroever, we point out that an upper bound on the volume of $\Pi(K)$ of a centrally symmetric $n$-dimensional convex body of volume $1$ is at least $2^n (9/8)^{\lfloor n/3\rfloor}$.
Martin Henk
. NOTE ON PROJECTION BODIES OF ZONOTOPES WITH ${n+1}$ GENERATORS[J]. Acta mathematica scientia, Series B, 2025
, 45(1)
: 96
-103
.
DOI: 10.1007/s10473-025-0107-9
[1] Brannen N S. Volumes of projection bodies. Mathematika, 1996, 43(2): 255-264
[2] Brannen N S. Three-dimensional projection bodies. Adv Geom, 2005, 5(1): 1-13
[3] Gardner R J. Geometric Tomography. New York: Cambridge University Press, 2006
[4] Gruber P M. Convex and Discrete Geometry. Berlin: Springer, 2007
[5] Ludwig M. Projection bodies and valuations. Advances in Mathematics, 2002, 172(2): 158-168
[6] Lutwak E. Centroid bodies and dual mixed volumes. Proc London Math Soc, 1990, S3-60(2): 365-391
[7] Lutwak E. Inequalities for mixed projection bodies. Transactions of the American Mathematical Society, 1993, 339(2): 901-916
[8] Lutwak E, Yang D, Zhang G Y. A new affine invariant for polytopes and Schneider's projection problem. Transactions of the American Mathematical Society, 2001, 353(5): 1767-1779
[9] Matheron G. Random Sets and Integral Geometry. Hoboken, NJ: John Wiley & Sons, 1975
[10] Petty C M. Projection bodies// Proc Colloquium on Convexity (Copenhagen, 1965), 1967: 234-241
[11] Saroglou C. Volumes of projection bodies of some classes of convex bodies. Mathematika, 2011, 57(2): 329-353
[12] Schneider R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 2014
[13] Shephard G C. Combinatorial properties of associated zonotopes. Can J Math, 1974, 26: 302-321
[14] Weil W. Kontinuierliche Linearkombination von Strecken. Mathematische Zeitschrift, 1976, 148: 71-84
