We study the set SNA(M, Y ) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space F (M ) over M , and show that all of them actually provide the norm density of SNA(M, Y ) in the space of all Lipschitz maps from M to any Banach space Y . Next, we prove that SNA(M, R) is weakly sequentially dense in the space of all Lipschitz functions for all metric spaces M . Finally, we show that the norm of the bidual space of F (M ) is octahedral provided the metric space M is discrete but not uniformly discrete or M is infinite.J. Lindenstrauss extended such study to general linear operators, showed that this is not always possible, and also gave positive results. If we say that a Banach space X has (Lindenstrauss) property A when NA(X, Y ) = L(X, Y ) for every Banach space Y , it is shown in [36] that reflexive spaces have property A. This result was extended by J. Bourgain [9] showing that Banach spaces X with the RNP also have Lindenstrauss property A, and he also provided a somehow reciprocal result. We refer the interested reader to the survey paper [3] for a detailed account on norm attaining linear operators.

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M, Y ) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M, Y ) has the Lip-BPB property for every Banach space Y . We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M, R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.Recently, the problem of deciding for which metric spaces M the set LipSNA(M, Y ) is (norm) dense in Lip 0 (M, Y ) has been studied. We refer the reader to [11], [16], [18], and [21], as references for this study. Let us collect here some of the known results. First, the density does not always hold, as in [21, Example 2.1] it is shown that LipSNA([0, 1], R) is not dense in Lip 0 ([0, 1], R). In fact, this can be generalized to length spaces [11, Theorem 2.2]. On the other hand, it is obvious that LipSNA(M, Y ) = Lip 0 (M, Y ) when M is finite (actually, this fact characterizes finiteness of M ). Besides, LipSNA(M, Y ) is dense in Lip 0 (M, Y ) for every Banach space Y when M is uniformly discrete, or M is countable and compact, or M is a compact Hölder metric space (i.e. M = (N, d θ ) for some metric space (N, d) and 0 < θ < 1), but

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We study the stability behavior of the Bishop-Phelps-Bollobás property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobás property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very close to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of Γ -flat operators and ACK structure. We get sharper results for the case of Lipschitz compact maps. The behavior of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above for the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps.

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We study the density of the set SNA(M, Y ) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which strongly attain their norm (i.e. the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that SNA(T, Y ) is not dense in Lip 0 (T, Y ) for any Banach space Y , where T denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e. every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold. Next, we construct metric spaces M satisfying that SNA(M, Y ) is dense in Lip 0 (M, Y ) regardless Y but which contains an isometric copy of [0, 1] and so the Lipschitz-free space F (M ) fails the Radon-Nikodým property, answering in the negative a posed question. Furthermore, an example M can be produced failing all the previously known sufficient conditions to get the density of strongly norm attaining Lipschitz maps. Finally, among other applications, we prove that given a compact metric M which does not contains any isometric copy of [0, 1] and a Banach space Y , if SNA(M, Y ) is dense, then SNA(M, Y ) actually contains an open dense subset and B F (M ) = co(str-exp B F (M ) ). Further, we show that if M is a boundedly compact metric space for which SNA(M, R) is dense in Lip 0 (M, R), then the unit ball of the Lipschitz-free space on M is the closed convex hull of its strongly exposed points.

We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite spherically symmetric trees do not exhibit pre-cutoff; this conclusion also holds for continuous-time simple random walks. This answers a question recently proposed by Gantert, Nestoridi, and Schmid. Finally, we study the stability of this result under rough isometries.

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