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It is usually considered that the solution of isoperimetric problems is only to be achieved through the medium of the more or less advanced technique of variation calculus. The following theorem is derived using only the notions of pure geometry and elementary calculus, and is interesting since the method of proof is easily extended to several other problems of the same type.

We prove best possible estimates on the length of level lines of solutions of quasilinear elliptic partial differential equations in plane annuli. This result generalizes theorems by M. Longinetti and by P. Laurence.

We present a randomised polynomial time algorithm for approximating the volume of a convex body K in n-dimensional Euclidean space. The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within K.

Publisher Summary This chapter discusses the Fourier series and the isoperimetric problem. The isoperimetric problem is to find the closed curve of a given arc length that encloses the greatest area. The arc length, s, must be measured in a fixed direction from any fixed point on a given curve. Then the coordinates of the general point on the curve can be expressed parametrically in terms of s. Because the curve is closed and has arc length l, both x and y are periodic functions of s with period l. This reasoning shows that the circle is better than any polygon. It is clear, though, that arguments of this type will not solve problem. The restrictions imposed are not really significant, although they may appear so.

The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g. [3] and [10]. When this upper bound is non-positive, the second named author has shown an isoperimetric inequality for such domains, see [11]. This result again gives the comparison result for volumes alluded to above together with a characterization of the totally geodesic submanifolds of hyperbolic space forms. In the present paper we find a corresponding sharp isoperimetric inequality for minimal submanifolds in spaces with sectional curvatures bounded from above by any constant. As a corollary we find again a characterization of the totally geodesic submanifolds of spherical space forms.

The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.

A short proof is given for a sharpened form of the isoperimetric inequality for curves on minimal surfaces. By following a line of development used by Sachs [7] in treating inequalities for plane curves, one can give an economical formulation to the proof of the isoperimetric theorem for curves on minimal surfaces. Let C be a smooth simple closed curve in Euclidean n-space, where n > 2. In what follows we shall assume, as can be achieved by a translation, that the centroid of arc length of C is at the origin. Hence, if x is the position vector, we assume

1. We shall consider here the isoperimetric problemfor convex regions in the plane and its extension to more general sets. Part I is completely elementary as regards the methods used and presupposes next to nothing on the part of the reader. In it we prove the existence and uniqueness of the solution to the isoperimetric problem in the plane, restricted to convex regions. The uniqueness of a solution, if one exists, can quite easily be extended to regions bounded by a Jordan curve, but the existence becomes a little more difficult to show. In the second part we consider the existence proof for these more general regions, and al-so formulate and prove a very general isoperimetric theorem valid for arbitrary measurable sets in the plane. This part presupposes a very little knowledge of point set topology and measure theory, and a few statements are left unproved. Some of the places whiere tile words "clearly" or "obviously" are used are admittedly actual gaps in the exposition. Tle author feels, however, that these gaps can be easily filled in by the intelligent reader by straightforward, if perhaps long, reasoning.

The slow progress in the theory of Finsler spaces as compared to Riemann spaces is partly due to lack of information regarding the corresponding local, that is the Minkowskian, geometry. Those Minkowskian features will contribute most to an understanding of Finsler spaces which are not merely verbal generalizations of known euclidean statements.' The purpose of the present note was originally only to show that the isoperimetric problem (for any dimension) in Minkowski spaces leads to such a feature. It turned out, however, that the plane problem can be solved in a general form-no longer significant for Finsler spaces-and then exhibits a phenomenon which is of interest for the theory of isoperimetric problems in the calculus of variations. The result seems to indicate that the standard methods may have followed too closely the pattern of the fixed endpoint problem. For that reason the plane case is here presented separately. The following are the results: let F(x,,y)be continuous, positive for x, y # 0, and positive homogeneous of order 1. The problem, to find among all simple closed curves x (t), y (t) with a given orientation and a given Minkowski length L = 5 F (x, y) dt one which bounds the greatest (euclidean) area, has a unique solution (up to translations) no matter whether the indicatrix C : F(x, y) =1 is convex or not. For non-convex C t-he solution is the same as for the boundary 0 of the convex closure of C as indicatrix and is homothetic to the polar reciprocal (figuratrix) of 0 with respect to the unit circle rotated through ? 7r/2. For Finsler spaces only the case is of interest where C is convex and has the origin as center. Since Finsler or Minkowski area differs from the euclidean area by a constant factor, the solution of the Minkovskian isoperimetric problem is the same as for the above problem. In intrinsic Minkowskian terms it may be described as the curve of length L for which as-new

We first investigate whether for given convex domains K 0 , K 1 in the Euclidean plane, for any rotation α, there is a translation x so that x + α K 1 ⊂ K 0 or x + α K 1 ⊃ K 0 . Then, we estimate the mixed isoperimetric deficit Δ 2 ( K 0 , K 1 ) of domains K 0 and K 1 via the known kinematic formulas of Poincaré and Blaschke in integral geometry. We obtain the sufficient condition for domain K 0 to contain, or to be contained in, convex domain x + α K 1 . Finally, we obtain the mixed isoperimetric inequality and some Bonnesen-style mixed inequalities. Those Bonnesen-style mixed inequalities obtained are the known Bonnesen-style inequalities if one of the domains is a disc. As a direct consequence, we obtain the strengthened Bonnesen isoperimetric inequality. MSC:52A10, 52A22.

We establish the equivalence of some affine isoperimetric inequalities which include the -Petty projection inequality, the -Busemann-Petty centroid inequality, the "dual" -Petty projection inequality, and the "dual" -Busemann-Petty inequality. We also establish the equivalence of an affine isoperimetric inequality and its inclusion version for -John ellipsoids.

In this paper, we obtain some Bonnesen-style Minkowski inequalities of mixed volumes of convex bodies K and L in the Euclidean space R n . Let L be the unit ball; we get some better Bonnesen-style isoperimetric inequalities than Dinghas’s result for n ≥ 3 . MSC:52A20, 52A40.