{"id":1668,"date":"2013-10-30T06:48:12","date_gmt":"2013-10-30T11:48:12","guid":{"rendered":"http:\/\/sites.williams.edu\/Morgan\/?p=1668"},"modified":"2013-11-01T06:27:30","modified_gmt":"2013-11-01T11:27:30","slug":"sharp-isoperimetric-bounds-for-convex-bodies","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2013\/10\/30\/sharp-isoperimetric-bounds-for-convex-bodies\/","title":{"rendered":"Sharp Isoperimetric Bounds for Convex Bodies"},"content":{"rendered":"

Emanuel Milman [M2, 2013] provides very general sharp lower bounds on perimeter to enclose prescribed volume, including the following special case of convex bodies in R<\/b>n<\/sup><\/i>. Here we give an alternative proof for that special case as suggested by Milman [M2, \u00a77.2]. For sharp upper bounds see my post<\/a>\u00a0[Mo1] on the Convex Body Isoperimetric Conjecture.<\/p>\n

\u00a0<\/b>2.1. Theorem\u00a0<\/b>[M2, Cor. 1.4, Case\u00a0p<\/i>\u00a0= 0].<\/b>\u00a0Let K be an open convex body in\u00a0R<\/b>n<\/sup>\u00a0(n\u22652) of volume V and diameter d. Let R be a subset of K with volume sV (0 < s \u2264 1\/2). Then the area A of the boundary of R in K satisfies:<\/i><\/p>\n

A<\/em> > (V<\/em>\/d<\/em>)An<\/sub>(s)<\/i> \u2265 (V<\/em>\/d<\/em>)A<\/i>\u221e<\/sub>(s)<\/i>,<\/p>\n

where<\/i><\/p>\n

An<\/sub>(s) = n<\/i> min {[s<\/em>(b<\/em>+1)n<\/sup><\/i> + (1-s<\/em>)b<\/em>n<\/sup><\/i>](n<\/em>-1)\/n<\/em><\/sup><\/i>\/[(b<\/em>+1)n<\/sup><\/i>–b<\/em>n<\/sup><\/i>] : b<\/em> \u2265 0},<\/p>\n

A<\/i>\u221e<\/sub>(s) = <\/i>min {[sa<\/em>en<\/sup><\/i>\u00a0+ (1-s<\/em>)a<\/i>]\/[ea<\/sup><\/i>-1] :\u00a0a<\/em>\u00a0\u2265 0},<\/p>\n

In particular, for n = 2 (where A denotes length and V denotes area),<\/i><\/p>\n

A<\/em> > (2V<\/em>\/d<\/em>)\u221a(s<\/em>(1-s<\/em>)).<\/p>\n

The lower bound An<\/sub>(s) is nonincreasing in n and sharp for every s and d (as V approaches 0). See Figure 1.<\/i><\/p>\n


\n<\/a> \"3D,E\"<\/a> \"8D,E\"<\/a> \"100D,E\"<\/a>\"\u221eD,E\"<\/a><\/p>\n

Figure 1. Graphs of the sharp lower bounds An<\/sub>(s)<\/i> for n<\/i> = 3, 8, 100, \u221e. These bounds are concave and decreasing in n<\/i>.<\/p>\n

Remarks. <\/i>One checks that A<\/i>\u221e<\/sub>(1\/2) = 1. Considering b<\/i>\u21920 shows that An<\/sub><\/i>(1\/2) \u2264 1 and hence must equal 1\/2. Since the quantities minimized are concave, the minima An<\/sub><\/i>(s<\/i>) and A<\/i>\u221e<\/sub>(s<\/i>) are concave. In particular, A<\/i>\u221e<\/sub>(s<\/i>) \u2265 2s<\/i>, the previous result of Dyer and Frieze [DF, \u00a74.3, 1991], with subtle refinements by Kannan et al.<\/i> [KLS, 1995], Bobkov [B], Milman [M1], and Steinerberger [S, 2013]. The earlier estimate A<\/i> \u2265 sV<\/i>\/d<\/i> [KK, Thm. 2, 1991] is relatively trivial modulo a little geometric measure theory. Barthe [Ba, Prop. 11] provided the excellent but not sharp lower bound (2V<\/i>\/d<\/i>0<\/sub>)(\u221as<\/i> + \u221a(1\u2013<\/i>s<\/i>)\u2013<\/i>1) if K<\/i> is contained in a ball of diameter d<\/i>0<\/sub>.<\/p>\n

Proof.<\/i> By scaling we may assume that d<\/i> = 1. We apply repeated subdivision after Payne-Weinberger and Lov\u00e1sz-Simonovits ([PW], [LS, \u00a72], cited by Milman [M3, \u00a77.2]) as described in Gromov [G, 3.5.27], which can be consulted for further detail. Remove a small \u03b5\u00a0neighborhood of \u2202R, <\/i>of volume \u03b3V<\/i>, leaving parts of R<\/i> and K<\/i>\u2013<\/i>R<\/i> of volumes \u03b1V<\/i> and \u03b2V<\/i> separated by distance 2\u03b5. Of course \u03b1\u00a0+ \u03b2\u00a0+ \u03b3\u00a0= 1. By the Ham-Sandwich Theorem there is a hyperplane with unit normal in any prescribed circle of directions that cuts both parts in half. Take the half K<\/i>2<\/sub> of K<\/i> with smaller total volume, with parts of R<\/i> and K<\/i>–R<\/i> of volumes \u03b1V<\/i>\/2 and \u03b2V<\/i>\/2, separated by distance at least 2\u03b5\u00a0by a region occupying volume \u03b32<\/sub>V<\/i>\/2 \u2264 \u03b3V\/2. Keep repeating. One can choose the hyperplanes so that the limit is a line segment, of length at most 1, with an inherited density, normalized to unit mass, and two parts of mass \u03b1\u221e<\/sub> and \u03b2\u221e <\/sub>in the same ratio as \u03b1\u00a0and \u03b2\u00a0separated by distance at least 2\u03b5 by set of mass \u03b3\u221e<\/sub> = 1\u2013<\/i>\u03b1\u221e<\/sub>\u2013<\/i>\u03b2\u221e<\/sub> \u2264 \u03b3. The 1\/n<\/i> power of such a density must be concave. The minimum is attained by an interval on a segment of length 1 with a density of the form (a<\/i>(x<\/i>+b<\/i>))n<\/sup><\/i> on [0, 1] or equivalently (ax<\/i>)n<\/sup><\/i> on [b<\/i>, b<\/i>+1], with unit mass.<\/p>\n

As\u00a0\u03b5\u21920, \u03b3V<\/i>\/2\u03b5 converges to the area (Minkowski content) A<\/i> of \u2202R<\/i> and \u03b1\u00a0and \u03b2\u00a0converge to s<\/i> and 1\u2013s<\/i>. Meanwhile the lower bound on \u03b3\/2\u03b5 \u2265 \u03b3\u221e<\/sub>\/2\u03b5 subconverges to the mass of a point separating the unit segment into pieces of mass s<\/i> and 1\u2013<\/i>s<\/i>, the minimum given by (ac<\/i>)n<\/sup><\/i>, where the weighted measure of [b<\/i>, c<\/i>] is s<\/i> and the weighted measure of [c<\/i>, b<\/i>+1] is 1-s<\/i>. Using these two constraints to eliminate the parameters a<\/i> and c<\/i> yields the first lower bound An<\/sub><\/i>. By construction An<\/sub><\/i> is nonincreasing in n<\/i>. Since for small V<\/i> the weighted segments are approximated by thin convex bodies or “needles”, An<\/sub><\/i> is a sharp lower bound. The inequality is strict, because not all such needles can have unit diameter. For the case n<\/i> = 2, one can compute A<\/i>2<\/sub>(s<\/i>) as stated.<\/p>\n

To obtain the weaker bound A<\/i>\u221e<\/sub>(s<\/i>), note that all of these densities are log concave, for which the lower bound is given by eax<\/sup><\/i> on [b<\/i>, b<\/i>+1]. This time it is easier to use the constraints to eliminate the parameters b<\/i> and c<\/i>, yielding the second, weaker lower bound A<\/i>\u221e<\/sub>(s<\/i>).<\/p>\n

Remark.<\/i> Choe et al. ([ChGR]; see also [Mo3]) provide an isoperimetric inequality in the complement of a convex body.<\/p>\n

Remark<\/em>. Geometric measure theory (see e.g. [Mo2]) provides inside an open convex body in R<\/b>n<\/sup><\/i> a region of given volume fraction and minimum perimeter. The interface is a constant-mean-curvature hypersurface, smooth except possibly for a singular set of Hausdorff dimension at most n<\/i>\u2013<\/i>8. If the convex body has a smooth boundary, then the interface is smooth at the boundary.<\/p>\n

 <\/p>\n

References<\/b><\/p>\n

[Ba] \u00a0F. Barthe, Log-concave and spherical models in isoperimetry, Geom. Funct. Anal. 12 (2002), 32-55.<\/p>\n

[Bo] \u00a0S. Bobkov, On isoperimetric constants for log-concave probability distributions, Geometric Aspects of Functional Analysis, Israel Seminar 2004-2005, Lectures Notes in Math. 1910, Springer, Berlin, 2007, 81-88.<\/p>\n

[ChGR] \u00a0Jaigyoung Choe, Mohammad Ghomi, and Manuel Ritor\u00e9, The relative isoperimetric inequality outside convex domains in R<\/b>n<\/sup><\/i>. Calc. Var. Partial Differential Equations 29 (2007), 421-429.<\/p>\n

[DF] \u00a0Martin Dyer and Alan Frieze, Computing the volume of convex bodies: a case where randomness provably helps, Proc. Symp. Applied Math 44 (1991), 123-169.<\/p>\n

[G] \u00a0Misha Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Math. 152, Birkh\u00e4user, second printing, 2001.<\/p>\n

[KLS] \u00a0Ravi Kannan, L\u00e1szl\u00f3 Lov\u00e1sz, and Mikl\u00f3s Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), 541\u2013559.<\/p>\n

[KK] \u00a0Alexander Karzanov and Leonid Khachiyan, On the conductance of order Markov chains, Order 8 (1991), 7-15.<\/p>\n

[LS] \u00a0L\u00e1szl\u00f3 Lov\u00e1sz, and Mikl\u00f3s Simonovits, The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume,\u00a0 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), 346\u2013354, IEEE Comput. Soc. Press, Los Alamitos, CA, 1990.<\/p>\n

[M1] \u00a0Emanuel Milman, Isoperimetric bounds on convex manifolds, Christian Houdr\u00e9, Michel Ledoux, Emanuel Milman, and Mario Milman, eds., Concentration, Functional Inequalities and Isoperimetry, Contemporary Math. 545 (2011), 195-208.<\/p>\n

[M2] \u00a0Emanuel Milman, Sharp isoperimetric inequalities and model spaces of curvature-dimension-diameter condition, http:\/\/arxiv.org\/abs\/1108.4609v1<\/a>.<\/p>\n

[Mo1] \u00a0Frank Morgan, Convex body isoperimetric conjecture, Morgan blog<\/a>.<\/p>\n

[Mo2] \u00a0Frank Morgan, Geometric Measure Theory: a Beginner’s Guide, Academic Press, 4th<\/sup> edition, 2009.<\/p>\n

[Mo3] \u00a0Frank Morgan, Isoperimetric inequality in complement of mean convex set fails at Banff, Morgan blog<\/a>.<\/p>\n

[PW] \u00a0L. E. Payne and H. F. Weinberger, An optimal Poincar\u00e9 inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286\u2013292.<\/p>\n

[S] \u00a0Stefan Steinerberger,\u00a0The optimal L<\/i>1<\/sup>-Poincar\u00e9 inequality and some isoperimetric applications, http:\/\/arxiv.org\/abs\/1309.6211<\/a><\/p>\n

 <\/p>\n

 <\/p>\n


\n<\/b><\/p>\n

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Emanuel Milman [M2, 2013] provides very general sharp lower bounds on perimeter to enclose prescribed volume, including the following special case of convex bodies in Rn. Here we give an alternative proof for that special case as suggested by Milman [M2, \u00a77.2]. For sharp upper bounds see my post\u00a0[Mo1] on the Convex Body Isoperimetric Conjecture.<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-1668","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1668","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=1668"}],"version-history":[{"count":16,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1668\/revisions"}],"predecessor-version":[{"id":1678,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1668\/revisions\/1678"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=1668"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=1668"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=1668"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}