We consider a market with proportional transaction costs and want to hedge a claim by trading in the underlying assets. The superhedging problem is to find the set of d-dimendional vectors of initial capital that allow to superhedge the claim. We will show that in analogy to the frictionless case, the superhedging price in a market with proportional transaction costs is a (set-valued) coherent risk measure, where the supremum in the dual representation is taken w.r.t. the set of equivalent martingale measures. To do so, we extend the notion of set-valued risk measure to the case of random solvency cones. Connections to recent results about efficient use of capital when there are multiple eligible assets are drawn. When starting with a vector of initial capital that does not allow to superhedge, a shortfall at maturity is possible. For an investor who finds a hedging error that is 'small enough' still acceptable, good-deal-bounds under transaction costs can be defined.