# PROPERTIES OF MINIMIZERS OF AVERAGE-DISTANCE .properties of minimizers of average-distance problem

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PROPERTIES OF MINIMIZERS OF AVERAGE-DISTANCE PROBLEM VIADISCRETE APPROXIMATION OF MEASURES

XIN YANG LU AND DEJAN SLEPCEV

ABSTRACT. Given a finite measure with compact support, and > 0, the average-distance problem, in the penalized formulation, is to minimize

(0.1) E() :=Rdd(x,)d(x) + H1(),

among pathwise connected, closed sets, . Here d(x,) is the distance from a pointto a set and H1 is the 1-Hausdorff measure. In a sense the problem is to find a one-dimensional measure that best approximates . It is known that the minimizer is topo-logically a tree whose branches are rectifiable curves. The branches may not be C1, evenfor measures with smooth density. Here we show a result on weak second-order reg-ularity of branches. Namely we show that arc-length-parameterized branches have BVderivatives and provide a priori estimates on the BV norm.

The technique we use is to approximate the measure , in the weak- topology of mea-sures, by discrete measures. Such approximation is also relevant for numerical compu-tations. We prove the stability of the minimizers in appropriate spaces and also comparethe topologies of the minimizers corresponding to the approximations with the mini-mizer corresponding to .

Keywords. nonlocal variational problem, average-distance problem, regularityClassification. 49Q20, 49K10, 49Q10, 35B65

1. INTRODUCTION

The average-distance problem was introduced by Buttazzo, Oudet and Stepanov in[1]. In the penalized formulation the problem takes the form

Problem 1.1. For d 2, given a finite compactly supported measure , and > 0, minimize

(1.1) E() =Rdd(x,)d(x) + H1()

among pathwise connected, closed sets .

We define the set of admissible candidates by

A := { Rd : compact, pathwise connected, andH1()

2 XIN YANG LU AND DEJAN SLEPCEV

For the future reference we denote the part of the energy functional which measures theapproximation error by

F() :=

Rdd(x,)d.

This problem has many applications: a simple one is is the distribution of passengers, is a transport network.

In this case F() is the average distance of passengers from the network, and H1()is the cost to build such network. Minimizing E is determining the network with bestserves the passengers, under cost considerations.

An alternative interpretation is found in cloud data systems: is the distribution of data points, is a one dimension object which approximates the data.

In this case F() is the approximation error, while H1() is the cost associated to thecomplexity of such representation. Minimizing E is determining the best approxima-tion, accounting for the complexity.

Existence of minimizers, follows from theorems of Blaschke and Goab, [1]. Geomet-ric properties of minimizers were investigated by Buttazzo, Oudet and Stepanov [1],Buttazzo and Stepanov [2, 3] (mainly in two dimensions) and by Paolini and Stepanov[9] (for higher dimensions). Let us also add that Lemenant [6] provides an excellent re-view of the current knowledge about average-distance problem. It has been proven thatminimizers are topologically trees with rectifiable branches. Furthermore no more thanthree branches can meet at a point. Regarding the regularity of branches the followingresults are available:

In [1] it has been proven that minimizers are Ahlfors regular in two dimensionaldomains. The result was extended to higher dimensional domains in [9], In [2] it has been proven that minimizers are union of Lipschitz curves; for the

penalized problem, it is easy to prove that this union if finite (e.g. using Theorem3.4). In [12] it has been proven that regularity beyond C1,1 cannot be expected in the

general case. In [10] an explicit counterexample to C1 regularity for minimizers has been con-

structed.We recall that in two dimensional domains, in the constrained formulation

minH1()L

F(),

in [11] it has been proven that given a minimizer , (which is finite union of Lipschitzcurves {i}Ni=1, as proven in [2]), if (part of) j satisfies a geometric condition then it is

PROPERTIES OF MINIMIZERS OF AVERAGE-DISTANCE PROBLEM 3

is C1,1. Moreover, in [7] which studies the average-distance problem in two dimensionsprovides "tilt estimates" which are related to our regularity estimates.

This paper aims to extend such regularity results, and prove that minimizers are finiteunions of Lipschitz curves with BV derivatives. More precisely we show that given anarbitrary nonnegative finite measure with compact support, , and > 0, any solution argmin E is finite union of Lipschitz curves {k}Nk=1 (without loss of generalityassume that all k are arc-length parameterized), such that for any k = 1, . . . , N it holds

(1.2)k

kTV 1

|(Rd)|,

where TV denotes the total variation.In other words we provide an estimate on the total curvature of the curves that make

up , where the curvature, = k is understood as the signed measure. The fact the thetotal mass (times 1/) bounds the curvature is not surprising. To motivate it let us as-sume that the minimizer is a single smooth curve and that is absolutely continuouswith respect to Lebesgue measure. Let be the projection onto (it is known that has unique value almost everywhere). Then the first variation for the Problem 1.1 givesthat for any smooth vector field v : Rd supported away from the endpoints of

vdH1() = 1

Rd

(x (x))|x (x)|

v(((x))d(x).

Taking supremum over all v, as above, with |v| 1 implies (1.2).The difficulties one faces in applying the reasoning above are that is neither regular

nor a curve and that is not assumed to be absolutely continuous with respect to theLebesgue measure.

The approach used is based on approximating a measure by a sequence discretemeasures {n}

, and analyzing the minimizers of Problem 1.1 with replaced by n.

In addition to the estimate on the BV norm we prove a topological relation betweenminimizers of En and minimizers of E

, with > 0 arbitrarily given.

This paper is structured as follows: In Section 2 we recall the known results on the average-distance problem, intro-

duce the discrete approximations and prove a couple of preliminary results. In Section 3 we prove an upper bound on the number endpoints of minimizers

and analyze the behavior of endpoints in the approximation process. In particu-lar we prove in Theorem 3.4 that if n are discrete approximations of and n areminimizers of En which converge to , a minimizer of E

, then each endpoint

of is a limit of endpoints of n. Section 4 is devoted to the topological comparison between minimizers for the

approximate problem and the minimizer corresponding to .

4 XIN YANG LU AND DEJAN SLEPCEV

In Section 5 we prove prove that minimizers of the average-distance problem arefinite union of Lipschitz curves withBV derivatives, and prove a priori estimateson BV norms in Theorem 5.1.

2. PRELIMINARY RESULTS

We restrict our attention to probability measures, , purely for notational simplicity.Given a probability measure and a compact set , we need to find a "projection" of themeasure onto . The issue is that for x Rd, the minimizer of |x y| over y isin general not unique. In fact it has been proven by the authors in [8] that the ridge of (points having nonunique projection on ) is anHd1-rectifiable set. If is absolutelycontinuous with respect to the Lebesgue measure, Ld, then the ridge is -negligibleand thus the projection : x 7 argminy |x y| is unique -a.e. and one can define theprojection of onto by = ] (the push-forward of the measure).

However if is not absolutely continuous with respect to Lebesgue measure thenmore care is needed.

Lemma 2.1. Let be a probability measure and a compact set. There exists a probabilitymeasure on Rd such that the first marginal of is (that is (A Rd) = (A) for anyBorel set A) and that for -a.e. (x, y), |x y| = minz |x z|.

We define , the projection of onto , to be the second marginal of . While isin general not unique, from the outset we choose one and the corresponding . Allstatements we make hold for any chosen.

Proof. Let P be the set of Borel probability measures on . Consider the functional 7 dW (, ) on P, where dW is the Wasserstein distance. Since is compact, Pis sequentially compact with respect to weak- convergence of measures. Given thatdW (, ) is continuous on P with respect to the weak- convergence of measures weconclude that there exists P minimizing the Wasserstein distance to . Let be theoptimal transportation plan (for the quadratic cost) between and . We claim that has the desired properties.

Since, by definition, the first marginal of is we only need to verify that for -a.e.(x, y), |x y| = minz |x z|. Assume that this is not the case, that is that there exists(x, y) supp and z such that |x y| > |x z|. Let = (|x y| |x z|)/3 andU = B(x, )(B(y, )). Since (x, y) supp , := (U) > 0. Let new = |U+(x,z)and let new the the second marginal of new. Then new P and dW (, new) < dW (, )which contradicts the fact that is a minimizer.

We introduce some notation and terminology: The measure defined after the statement of Lemma 2.1 can have atoms. For

simplicity for y we write (y) to mean ({y}).

PROPERTIES OF MINIMIZERS OF AVERAGE-DISTANCE PROBLEM 5

Sometimes it is important to emphasize which and the measure corre-sponds to. Then we write (,, A) for (A), where A is a measurable subset of. The order of a point y is defined to be the number of pathwise connected

components of \{y}. As we mentioned earlier if is

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