By Guy David

Approximately conversing, a $d$-dimensional subset of $\mathbfR^n$ is minimizing if arbitrary deformations of it (in an appropriate classification) can't reduce its $d$-dimensional quantity. For quasiminimizing units, one permits the mass to diminish, yet simply in a managed demeanour. To make this detailed we stick with Almgren's thought of 'restricted units' [{\textbold 2}]. Graphs of Lipschitz mappings $f\:\mathbfR^d \to \mathbfR^{n-d}$ are continually quasiminimizing, and Almgren confirmed that quasiminimizing units are rectifiable. the following we identify uniform rectifiability houses of quasiminimizing units, which supply a extra quantitative experience within which those units behave like Lipschitz graphs. (Almgren additionally confirmed enhanced smoothness homes lower than tighter quasiminimality conditions.)Quasiminimizing units can come up as minima of functionals with hugely abnormal 'coefficients'. For such functionals, one can't wish ordinarily to have even more within the approach of smoothness or constitution than uniform rectifiability, for purposes of bilipschitz invariance. (See additionally [{\textbold 9}].) One motivation for contemplating minimizers of functionals with abnormal coefficients comes from the next kind of query. believe that one is given a compact set $K$ with top bounds on its $d$-dimensional Hausdorff degree, and decrease bounds on its $d$-dimensional topology.What can one say concerning the constitution of $K$? To what quantity does it behave like a pleasant $d$-dimensional floor? A easy approach for facing this factor is to first change $K$ by way of a suite that's minimizing for a size of quantity that imposes a wide penalty on issues which lie outdoors of $K$. This results in a type of regularization of $K$, during which cusps and extremely scattered elements of $K$ are got rid of, yet with no including greater than a small quantity from the supplement of $K$. the implications for quasiminimizing units then result in uniform rectifiability homes of this regularization of $K$. to really produce minimizers of normal functionals it really is occasionally handy to paintings with (finite) discrete types. a pleasant function of uniform rectifiability is that it offers the way to have bounds that cooperate robustly with discrete approximations, and which live to tell the tale within the restrict because the discretization turns into finer and finer.

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27). 6) ) In other words. — ru1÷Q, and so (i + Of course we have a similar inequality for rne+ i and mg when F < i (assuming that the Qj's for F i were chosen in the same manner as for Qt+fl. This implies that rnjptrno (430) for (which satisfies p < I this time). 30), and the fact that j(F) 10 for all F 1. is small enough Note that the constants C'. C" here depend only on it and k If (depending only on n and k). 26), and we can repeat the process. 32) 4 LOCAL AIILF'ORS-REGULARITY for all i 0. )) Thus = 0 we conclude that W' (S fl In particular, cannot be centered on the support of Hd restricted to S.

The second reason is that if F is a bounded regular set, then it is very easy to derive the existence of a cubical patchwork for F from the analogous result for unbounded sets. To see this, consider the set F = F U F, where P is a d—plane whose distance to F is equal to R = diam F. 4). but with j now running through Z and F replaced with F. 2); for compatibility with and 15], let us take 1? 2) for {E3 } and F, there is a smallest integer no with j then Q is either wholly contained in F, or such that if Q E wholly contained in the d-plane P (and hence disjoint from F).

2). 19). 2). 3). Let Q be a cube contained in U which satisfies diam Q <8, and let j be any arbitrary positive integer (for which we shall make a particular choice later). 1 with E = S fl Q to get a Lipschitz mapping but we shall not use the quasiminimality of S directly with this choice of 0. 6), and /3 is a positive number which depends only on ri and d and which is about to be specified. 22) were not true. then we would get a lower bound for H"(S fl Q). ) Let T be any one of the d-dimensional cubes in Ajd(Q) of which S3,d(Q) is composed.