ARCHIMEDES' PRINCIPLE OF FLOTATION AND FLOATING BODIES: CONSTRUCTION, EXTENSIONS AND RELATED PROBLEMS

ARCHIMEDES' PRINCIPLE OF FLOTATION AND FLOATING BODIES: CONSTRUCTION, EXTENSIONS AND RELATED PROBLEMS

ARCHIMEDES' PRINCIPLE OF FLOTATION AND FLOATING BODIES: CONSTRUCTION, EXTENSIONS AND RELATED PROBLEMS

Abstract

References

Metrics

Comments

Recommended 10

Abstract

Cite this article

share this article

References

Related Articles 2

Metrics

Comments

Recommended 10

doi:10.1007/s10473-025-0119-5

Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (1): 237-256.doi: 10.1007/s10473-025-0119-5

Previous Articles Next Articles

Chunyan Liu¹, Elisabeth M. Werner², Deping Ye³, Ning Zhang⁴

1. School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430048, China;
2. Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA;
3. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada A1C 5S7;
4. School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China

Received:2024-09-22 Revised:2024-10-26 Published:2025-02-06
About author:Chunyan Liu, E-mail,: chunyanliu@whpu.edu.cn; Elisabeth M. Werner, E-mail,: elisabeth.werner@case.edu; Deping Ye, E-mail,: deping.ye@mun.ca; Ning Zhang, E-mail,: nzhang2@hust.edu.cn
Supported by:
Chunyan Liu's work was supported by the Research Funding of Wuhan Polytechnic University (2024RZ083). Elisabeth M. Werner's work was supported by the NSF grant DMS-2103482. Deping Ye's work was supported by an NSERC grant, Canada. Ning Zhang's work was supported by the NSF of China (11901217, 11971005).

CLC Number:

52A20

Chunyan Liu, Elisabeth M. Werner, Deping Ye, Ning Zhang. ARCHIMEDES' PRINCIPLE OF FLOTATION AND FLOATING BODIES: CONSTRUCTION, EXTENSIONS AND RELATED PROBLEMS[J].Acta mathematica scientia,Series B, 2025, 45(1): 237-256.

Trendmd

[1] Auerbach H. Sur un Problème de M. Ulam Concernant l'Equilibre des Corps Flottants. Stud Math, 1938, 7: 121-142
[2] Bárány I, Larman D G. Convex bodies, economic cap coverings, random polytopes. Mathematika, 1988, 35(2): 274-291
[3] Blaschke W. Vorlesung Über Differentialgeometrie II: Affine Differntialgeometrie. Berlin: Springer-Verlag, 1923
[4] Besau F, Werner E. The spherical convex floating body. Adv Math, 2016, 301: 867-901
[5] Besau F, Werner E. The floating body in real space forms. J Differential Geom, 2018, 110: 187-220
[6] Bracho J, Montejano L, Oliveros D. Carousels, Zindler curves and the floating body problem. Per Mat Hungarica, 2004, 2(49): 9-23
[7] Caglar U, Werner E. Divergence for

s

$s$ -concave and log-concave functions. Adv Math, 2014, 257: 219-247
[8] Caglar U, Werner E. Mixed

f

$f$ -divergence and inequalities for log-concave funtions. Proc London Math Soc, 2015, 110: 271-290
[9] Caglar U, Fradelizi M, Guédon O, et al. Functional versions of

L_{p}

$L_p$ -affine surface area and entropy inequalities. Int Math Res Not, 2016, 4: 1223-1250
[10] Dupin C. Application de géométrie et de méchanique. Paris, 1822
[11] Falconer K. Applications of a result on spherical integration to the theory of convex sets. Amer Math Monthly, 1983, 90: 690-69
[12] Florentin D, SchÜtt C, Werner E, Zhang N. Convex floating bodies of equilibrium. Proc Amer Math Soc, 2022, 150: 3037-3048
[13] Gruber P M. Asymptotic estimates for best and stepwise approximation of convex bodies II. Forum Math, 1993, 5: 521-538
[14] Hug D. Contributions to affine surface area. Manuscripta Math, 1996, 91: 283-301
[15] Huang H, Slomka B A. Approximations of convex bodies by measure-generated sets. Geom Dedicata, 2019, 200: 173-19
[16] Huang H, Slomka B A, Werner E. Ulam floating bodies. J London Math Soc, 2019, 100: 425-446
[17] Kiener K. Extremailtät von Ellipsoiden und die Faltungsungleichung von Sobolev. Arch Math, 1986, 46: 162-168
[18] Li B, SchÜtt C, Werner E. Floating functions. Israel J Math, 2019, 231: 181-210
[19] Liu C, Werner E, Ye D, Zhang N. Ulam floating functions. J Geom Anal, 2023, 33: 1-25
[20] Leichtweiss K. Über ein Formel Blaschkes zur Affinoberfäche. Studia Scient Math Hung, 1986, 21: 453-474
[21] Leichtweiss K. Zur Affinoberfläche konvexer Körper. Manuscripta Mathematica, 1986, 56: 429-464
[22] Ludwig M, Reitzner M. A characterization of affine surface area. Adv Math, 1999, 147: 138-172
[23] Ludwig M. General affine surface areas. Adv Math, 2010, 224: 2346-2360
[24] Lutwak E. Extended affine surface area. Adv Math, 1991, 85: 39-68
[25] Lutwak E. The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas. Adv Math, 1996, 118: 244-294
[26] McClure D, Vitale R. Polygonal approximation of plane convex bodies. J Math Anal Appl, 1975, 51: 326-358
[27] Meyer M, Werner E. The Santaló-regions of a convex body. Trans Amer Math Soc, 1998, 350: 4569-4591
[28] Meyer M, Werner E. On the

p

$p$ -affine surface area. Adv Math, 2000, 152: 288-313
[29] Petty C. Isoperimetric problems// Proc Conf Convexty and Combinatorial Geometry, University of Oklahoma, Norman, Okla. 1972: 26-41
[30] Ryabogin D. A negative answer to Ulam's problem 19 from the Scottish Book. Ann Math, 2022, 195: 1111-1150
[31] Ryabogin D. On bodies floating in equilibrium in every orientation. Geom Dedicata, 2023, 217: Art 70
[32] Schneider R. Functional equations connected with rotations and their geometric applications. Enseign Math, 1970, 16: 297-305
[33] Schneider R. Convex Bodies: The Brunn-Minkowski Theory, Cambridge: Cambridge Univ Press, 2014
[34] Schütt C, Werner E. The convex floating body. Math Scand, 1990, 66: 275-290
[35] SchÜtt C. The convex floating body and polyhedral approximation. Israel J Math, 1991, 73: 65-77
[36] Schütt C. On the affine surface area. Proc Am Math Soc, 1993, 118: 1213-1218
[37] SchÜtt C, Werner E. Polytopes with vertices chosen randomly from the boundary of a convex body// Geometric Aspects of Functional Analysis: lsrael Seminar. Berlin: Springer, 2003: 241-422
[38] Schütt C, Werner E. Surface bodies and

p

$p$ -affine surface area. Adv in Math, 2004, 187: 98-145
[39] SchÜtt C, Werner E. Homothetic floating bodies. Geom Dedicata, 1994, 49: 335-348
[40] Schütt C, Werner E. Floating Bodies. Book in preparation, to be published by the AMS.
[41] Schmuckenschläger M. The distribution function of the convolution square of a convex symmetric body in

R^{n}

$\mathbb{R}^n$ . Israel J Math, 1992, 78: 309-334
[42] Stancu A. Two volume product inequalities and their applications. Canad math Bull, 2009, 52: 464-472
[43] Stancu A. The floating body problem. Bull London Math Soc, 2006, 38: 839-846
[44] Ulam S. A Collection of Mathematical Problems. New York: Interscience Publishers, 1960
[45] Wegner F. Floating bodies of equilibrium. Stud Appl Math, 2003, 111: 167-183
[46] Wegner F. Floating bodies in equilibrium in

2 D

$2D$ , the tire track problem and electrons in a parabolic magnetic fields. arXiv:physics/0701241v3
[47] Wegner F. Floating bodies of equilibrium in three dimensions. The central symmetric case. arXiv:0803.1043
[48] Werner E. The

p

$p$ -affine surface area and geometric interpretations. Rend Circ Math Palermo Serie II, 2002, 70: 367-382
[49] Werner E, Ye D. New

L_{p}

$L_p$ affine isoperimetric inequalities. Adv Math, 2008, 218: 762-780
[50] Werner E, Ye D. On the homothety conjecture. Indiana University Mathematics Journal, 2011, 60: 1-20
[51] Zhang G. Restricted chord projection and affine inequalities. Geom Dedicata, 1991, 39: 213-222

Viewed

Full text

Abstract

Just accepted	Online first	Issue

0	0	32

From	Others	local

Times	31	1
Rate	97%	3%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	Yinzheng Sun, Aifang Qu, Hairong Yuan. THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 37 -77 .
[2]	Jian CHEN. ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES[J]. Acta mathematica scientia,Series B, 2024, 44(2): 431 -444 .
[3]	Xiaoman Duan, Zhuangdan Guan. COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES (RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 78 -102 .
[4]	. PREFACE[J]. Acta mathematica scientia,Series B, 2025, 45(1): 1 -2 .
[5]	Martin Henk. NOTE ON PROJECTION BODIES OF ZONOTOPES WITH ${n+1}$ GENERATORS[J]. Acta mathematica scientia,Series B, 2025, 45(1): 96 -103 .
[6]	Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng. ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES[J]. Acta mathematica scientia,Series B, 2024, 44(3): 823 -838 .
[7]	Jianfei WANG, Yanhui ZHANG. THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$ p OF Cn[J]. Acta mathematica scientia,Series B, 2024, 44(3): 839 -850 .
[8]	Quanquan Yao, Peiyong Zhu. MEAN SENSITIVITY AND BANACH MEAN SENSITIVITY FOR LINEAR OPERATORS[J]. Acta mathematica scientia,Series B, 2024, 44(4): 1200 -1228 .
[9]	Xiaohui Li, Meiqi Liu, Chunyuan Deng. REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS[J]. Acta mathematica scientia,Series B, 2024, 44(4): 1229 -1243 .
[10]	Yu Li, Bing Wang. HEAT KERNEL ON RICCI SHRINKERS (II)^*[J]. Acta mathematica scientia,Series B, 2024, 44(5): 1639 -1695 .

Just accepted

Online first

Just accepted

Online first

[1]	Xiao Li, Jiazu Zhou. A FUNCTIONAL ORLICZ BUSEMANN-PETTY CENTROID INEQUALITY FOR LOG-CONCAVE FUNCTIONS [J]. Acta mathematica scientia,Series B, 2025, 45(1): 52-71.
[2]	M. Angeles Alfonseca, Dmitry Ryabogin, Alina Stancu, Vladyslav Yaskin. CHARACTERIZATIONS OF BALLS AND ELLIPSOIDS BY INFINITESIMAL HOMOTHETIC CONDITIONS [J]. Acta mathematica scientia,Series B, 2025, 45(1): 280-290.