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We study the geometry of $\mathcal{L}(X)_w,$ the space of all bounded linearoperators on a Banach space $X,$ endowed with the numerical radius norm,whenever the numerical radius defines a norm. We obtain the form of the extremepoints of the unit ball of the dual space of $\mathcal{L}(X)_w.$ Using thisstructure, we explore Birkhoff-James orthogonality, best approximation anddeduce distance formula in $\mathcal{L}(X)_w.$ A special attention is given tothe case of operators satisfying a notion of smoothness. Finally, we obtain anequivalence between Birkhoff-James orthogonality in $\mathcal{L}(X)_w$ and thatin $X.$

Extreme points of the unit ball of $\mathcal{L}(X)_w^*$ and best approximation in $\mathcal{L}(X)_w$