  We consider a class of solutions of the WDVV equation related to the special
systems of covectors (called $\vee$-systems) and show that the corresponding
logarithmic Frobenius structures can be naturally restricted to any
intersection of the corresponding hyperplanes. For the Coxeter arrangements the
corresponding structures are shown to be almost dual in Dubrovin's sense to the
Frobenius structures on the strata in the discriminants discussed by Strachan.
For the classical Coxeter systems this leads to the families of $\vee$-systems
from the earlier work by Chalykh and Veselov. For the exceptional Coxeter
systems we give the complete list of the corresponding $\vee$-systems. We
present also some new families of $\vee$-systems, which can not be obtained in
such a way from the Coxeter systems.
