If \(X\) is a Banach space, \(K\subset X^*\) a \(w^*\)-compact set, a set \(B\subset K\) is a boundary of \(K\) if every \(x\in X\) attains its maximum on \(K\) at some point of \(B\). A~trivial example of a boundary is \(K\) itself, less trivial is the set \(\text{ext} K\) of all extreme points of \(K\). If \(B\) is a boundary of a \(w^*\)-compact set \(K\), it holds that \(\overline{\text{co}}^{w^*}(B)=\overline{\text{co}}^{w^*}(K)\) and sometimes even \(\overline{\text{co}}(B)=\overline{\text{co}}^{w^*}(K)\). The aim of the paper is to study consequences of the fact that \(\overline{\text{co}}(B)\neq \overline{\text{co}}^{w^*}(K)\), namely, how to localize inside \(K\) (or even \(B\)) a copy of the basis of \(\ell_1(\mathfrak{c})\) and a so-called \(w^*\)-\(\mathbb N \)-family. An important tool for ``localization'' results is a computation of the distance of a vector \(\psi\in X^{**}\) to several spaces of Baire-1 functions. More precisely, if \(d>0\) and \(\sup\langle \psi, \overline{\text{co}}^{w^*} (K)\rangle>\sup \langle \psi, B\rangle +d\), then the number \(d\) is related to the distance of \(\psi\) to the space of Baire-1 bounded functions on \(K\) or the subspace of \(X^{**}\) consisting of Baire-1 functions on the dual unit ball \((B(X ^*),w^*)\). Another notion serving for finding copies of \(\ell_1(\mathfrak{c})\) is the notion of a \(w^*\)-\(\mathbb{N}\)-family. This is a bounded set \(A\subset X^*\) of \(\operatorname{width}(A)\geq d>0\) of the form \[ A=\{\eta_{M,N}: M,N \text{disjoint subsets of }\mathbb{N}\} \] for which there exist two sequences \(\{r_m: m\geq 1\}\subset \mathbb{R}\) and \(\{x_m: m\geq 1\}\subset B(X)\) such that, for every pair \(M,N\) of disjoint subsets of \(\mathbb{N}\), one has \[ \eta_{M,N}(x_n)\leq r_n,\quad \eta_{M,N}(x_m)\geq r_m+d,\quad m\in M, \;n\in N. \] A \(w^*\)-\(\mathbb{N}\)-family \(A\) contains a copy of the basis of \(\ell_1(\mathfrak{c})\) and the family \(\{x_m:m\geq 1\}\) is equivalent to the basis of \(\ell_1\). The index \(\operatorname{Width}(Y)\) of a set \(Y\subset X^*\) is defined as \[ \operatorname{Width}(A)=\sup\{d>0: \text{exists a \(w^*\)-\(\mathbb{N}\)-family \(A\subset Y\) with }\operatorname{width}(A)\geq d\}. \] Another index defined in the paper for a \(w^*\)-compact set \(K\subset X^*\) is \[ \operatorname{Bindex}(K)=\sup\{\text{dist} (\overline{\text{co}}^{w^*}(W),\overline{\text{co}}(B): W\subset K \text{ \(w^*\)-compact}, B\subset W\text{ a boundary of }W\} \] (here, \(\text{dist} (A,B)=\sup\{\text{dist}(a,B): a\in A\}\)). It is shown that, for a \(w^*\)-compact set \(H\subset X^*\), one has \(\operatorname{Width}(H)\leq \operatorname{Bindex}(H)\). If, moreover, \(H\) is convex and \(w^*\)-metrizable, it also holds that \(\operatorname{Bindex}(H)\leq 3 \operatorname{Width}(H)\). For general \(w^*\)-compact sets \(K\subset X^*\) a countable variant of \(\operatorname{Bindex}\) is introduced, namely \(\operatorname{Bindex}_c(K)\) is the supremum of \(\operatorname{Bindex}(i^*(K))\), where \(i^*\) is the adjoint operator of the canonical inclusion mapping \(i:Y\to X\) and \(Y\) is a separable subspace of \(X\). It is proved in the paper that the following statements are equivalent for a \(w^*\)-compact set \(K\subset X^*\): \(\operatorname{Width}(\overline{\text{co}}^{w^*}(K))=0\); \(\operatorname{Bindex}_c(\overline{\text{co}}^{w^*}(K))=0\); \(\operatorname{Width}(K)=0\); \(\operatorname{Bindex}_c(K)=0\); \(\overline{\text{co}}^\gamma(B)=\overline{\text{co}}^{w^*}(H)\) for every \(w^*\)-compact \(H\subset K\) and every boundary \(B\) of \(H\) (here, \(\gamma \) is the topology of the convergence on countable bounded subsets of \(X\)). There are counterexamples showing that, for a general boundary \(B\) of a \(w^*\)-compact set \(K\), the information \(\operatorname{Bindex}(K)>0\) does not imply that \(\operatorname{Width}(K)>0\). If a boundary is nicer from the point of view of descriptive set theory (either \(w^*\)-\(\mathcal{K}\)-analytic or \(w^*\)-countably determined), then more can be said. Among other results, it is proved that a Banach space \(X\) fails to have a copy of \(\ell_1(\mathfrak{c})\) if and only if, for every \(w^*\)-compact \(K\subset X^*\) and its every boundary \(B\), one has \(\overline{\text{co}}^\gamma(B)=\overline{\text{co}}^{w^*}(K)\), and this is the case if and only if, for every \(w^*\)-compact \(K\subset X^*\) and its every \(w^*\)-countably determined boundary \(B\), one has \(\overline{\text{co}}(B)=\overline{\text{co}}^{w^*}(K)\). If \(K\subset X^*\) is \(w^*\)-compact and \(B\subset K\) is a boundary which is either \(w^*\)-\(\mathcal{K}\)-analytic or \(B=\text{ext} K\), then \(B\) contains a \(w^*\)-\(\mathbb{N}\)-family if and only if \(K\) does if and only if \(\overline{\text{co}}^{w^*}(K)\) does. Further, \(B\) contains a copy of the basis of \(\ell_1(\mathfrak{c})\) if and only \(K\) does if and only if \(\overline{\text{co}}^{w^*}(K)\) does.