## On a version of the slicing problem for the surface area of convex bodies

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- by Silouanos Brazitikos and Dimitris-Marios Liakopoulos PDF
- Trans. Amer. Math. Soc.
**375**(2022), 5561-5586 Request permission

## Abstract:

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha _n$ depending (or not) on the dimension $n$ so that \begin{equation*} S(K)\leqslant \alpha _n|K|^{\frac {1}{n}}\max _{\xi \in S^{n-1}}S(K\cap \xi ^{\perp }), \end{equation*} where $S$ denotes surface area and $|\cdot |$ denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann–Petty problem with the slicing problem has an affirmative answer.## References

- Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman,
*Asymptotic geometric analysis. Part I*, Mathematical Surveys and Monographs, vol. 202, American Mathematical Society, Providence, RI, 2015. MR**3331351**, DOI 10.1090/surv/202 - Franck Barthe,
*An extremal property of the mean width of the simplex*, Math. Ann.**310**(1998), no. 4, 685–693. MR**1619740**, DOI 10.1007/s002080050166 - J. Bourgain,
*On the distribution of polynomials on high-dimensional convex sets*, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. MR**1122617**, DOI 10.1007/BFb0089219 - Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou,
*Geometry of isotropic convex bodies*, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR**3185453**, DOI 10.1090/surv/196 - Silouanos Brazitikos, Susanna Dann, Apostolos Giannopoulos, and Alexander Koldobsky,
*On the average volume of sections of convex bodies*, Israel J. Math.**222**(2017), no. 2, 921–947. MR**3722270**, DOI 10.1007/s11856-017-1561-4 - Yu. D. Burago and V. A. Zalgaller,
*Geometric inequalities*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR**936419**, DOI 10.1007/978-3-662-07441-1 - H. Busemann and C. M. Petty,
*Problems on convex bodies*, Math. Scand.**4**(1956), 88–94. MR**84791**, DOI 10.7146/math.scand.a-10457 - H. Busemann and E. G. Straus,
*Area and normality*, Pacific J. Math.**10**(1960), 35–72. MR**121767** - Giorgos Chasapis, Apostolos Giannopoulos, and Dimitris-Marios Liakopoulos,
*Estimates for measures of lower dimensional sections of convex bodies*, Adv. Math.**306**(2017), 880–904. MR**3581320**, DOI 10.1016/j.aim.2016.10.035 - Yuansi Chen,
*An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture*, Geom. Funct. Anal.**31**(2021), no. 1, 34–61. MR**4244847**, DOI 10.1007/s00039-021-00558-4 - Nikos Dafnis and Grigoris Paouris,
*Small ball probability estimates, $\psi _2$-behavior and the hyperplane conjecture*, J. Funct. Anal.**258**(2010), no. 6, 1933–1964. MR**2578460**, DOI 10.1016/j.jfa.2009.06.038 - Nikos Dafnis and Grigoris Paouris,
*Estimates for the affine and dual affine quermassintegrals of convex bodies*, Illinois J. Math.**56**(2012), no. 4, 1005–1021. MR**3231472** - Richard J. Gardner,
*Geometric tomography*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR**2251886**, DOI 10.1017/CBO9781107341029 - Apostolos Giannopoulos and Emanuel Milman,
*$M$-estimates for isotropic convex bodies and their $L_q$-centroid bodies*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 159–182. MR**3364687**, DOI 10.1007/978-3-319-09477-9_{1}3 - A. Giannopoulos, M. Hartzoulaki, and G. Paouris,
*On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body*, Proc. Amer. Math. Soc.**130**(2002), no. 8, 2403–2412. MR**1897466**, DOI 10.1090/S0002-9939-02-06329-3 - Apostolos Giannopoulos, Alexander Koldobsky, and Petros Valettas,
*Inequalities for the surface area of projections of convex bodies*, Canad. J. Math.**70**(2018), no. 4, 804–823. MR**3813513**, DOI 10.4153/CJM-2016-051-x - Eric L. Grinberg,
*Isoperimetric inequalities and identities for $k$-dimensional cross-sections of convex bodies*, Math. Ann.**291**(1991), no. 1, 75–86. MR**1125008**, DOI 10.1007/BF01445191 - R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch,
*Determining starlike bodies by the perimeters of their central sections*, preprint. - Zakhar Kabluchko and Dmitry Zaporozhets,
*Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls*, Trans. Amer. Math. Soc.**368**(2016), no. 12, 8873–8899. MR**3551592**, DOI 10.1090/tran/6628 - B. Klartag,
*On convex perturbations with a bounded isotropic constant*, Geom. Funct. Anal.**16**(2006), no. 6, 1274–1290. MR**2276540**, DOI 10.1007/s00039-006-0588-1 - B. Klartag and V. Milman,
*Rapid Steiner symmetrization of most of a convex body and the slicing problem*, Combin. Probab. Comput.**14**(2005), no. 5-6, 829–843. MR**2174659**, DOI 10.1017/S0963548305006899 - Alexander Koldobsky,
*Fourier analysis in convex geometry*, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. MR**2132704**, DOI 10.1090/surv/116 - A. Koldobsky,
*Stability and separation in volume comparison problems*, Math. Model. Nat. Phenom.**8**(2013), no. 1, 156–169. MR**3022986**, DOI 10.1051/mmnp/20138111 - Alexander Koldobsky,
*Stability inequalities for projections of convex bodies*, Discrete Comput. Geom.**57**(2017), no. 1, 152–163. MR**3589060**, DOI 10.1007/s00454-016-9844-9 - Alexander Koldobsky,
*A $\sqrt {n}$ estimate for measures of hyperplane sections of convex bodies*, Adv. Math.**254**(2014), 33–40. MR**3161089**, DOI 10.1016/j.aim.2013.12.029 - Alexander Koldobsky,
*Estimates for measures of sections of convex bodies*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 261–271. MR**3364691**, DOI 10.1007/978-3-319-09477-9_{1}7 - Alexander Koldobsky,
*Slicing inequalities for measures of convex bodies*, Adv. Math.**283**(2015), 473–488. MR**3383809**, DOI 10.1016/j.aim.2015.07.019 - A. Koldobsky,
*Private communication*. - Hermann König and Alexander Koldobsky,
*On the maximal perimeter of sections of the cube*, Adv. Math.**346**(2019), 773–804. MR**3914180**, DOI 10.1016/j.aim.2019.02.017 - Erwin Lutwak,
*Dual mixed volumes*, Pacific J. Math.**58**(1975), no. 2, 531–538. MR**380631** - V. D. Milman and A. Pajor,
*Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space*, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR**1008717**, DOI 10.1007/BFb0090049 - S. Myroshnychenko, K. Tatarko and V. Yaskin,
*Unique determination of ellipsoids by their dual volumes*, Int. Math. Res. Not. IMRN (to appear). - Robert Osserman,
*Bonnesen-style isoperimetric inequalities*, Amer. Math. Monthly**86**(1979), no. 1, 1–29. MR**519520**, DOI 10.2307/2320297 - C. Radhakrishna Rao,
*Linear statistical inference and its applications*, 2nd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1973. MR**0346957** - Igor Rivin,
*Surface area and other measures of ellipsoids*, Adv. in Appl. Math.**39**(2007), no. 4, 409–427. MR**2356429**, DOI 10.1016/j.aam.2006.08.009 - Anamaria Rusu,
*Determining starlike bodies by their curvature integrals*, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of South Carolina. MR**2711955** - Dmitry Ryabogin, Vlad Yaskin, and Artem Zvavitch,
*Harmonic analysis and uniqueness questions in convex geometry*, Recent advances in harmonic analysis and applications, Springer Proc. Math. Stat., vol. 25, Springer, New York, 2013, pp. 327–337. MR**3066896**, DOI 10.1007/978-1-4614-4565-4_{2}6 - Michael Schmuckenschläger,
*An extremal property of the regular simplex*, Convex geometric analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 199–202. MR**1665592** - Rolf Schneider,
*Convex bodies: the Brunn-Minkowski theory*, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR**3155183** - V. Yaskin,
*On perimeters of sections of convex polytopes*, J. Math. Anal. Appl.**371**(2010), no. 2, 447–453. MR**2670125**, DOI 10.1016/j.jmaa.2010.05.050

## Additional Information

**Silouanos Brazitikos**- Affiliation: Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis 157-84, Athens, Greece
- MR Author ID: 1059952
- Email: silouanb@math.uoa.gr
**Dimitris-Marios Liakopoulos**- Affiliation: Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis 157-84, Athens, Greece
- MR Author ID: 1189494
- Email: dliakop@math.uoa.gr
- Received by editor(s): July 5, 2021
- Received by editor(s) in revised form: December 25, 2021
- Published electronically: April 26, 2022
- Additional Notes: We acknowledge support by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant” (Project Number: 1849).
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5561-5586 - MSC (2020): Primary 52A20; Secondary 46B06, 52A40, 52A38, 52A23
- DOI: https://doi.org/10.1090/tran/8666
- MathSciNet review: 4469229