Risk tomography

New multivariate risk measures are introduced, suitable for optimal management of multidimensional assets. Risk is measured along lines through a given reference point in a multidimensional Euclidean space, and then maximum (minimum in financial planning) or mixture is taken with respect to lines lying in cones. We use VaR and CVaR as univariate risk measures but the construction allows for the use any of them. In some case numéraire is used to value the assets. Some of the new measures enjoy the coherence property for sums and also for composition, where assets are put together to form higher dimensional vectors. Numerical calculations of them are tractable as shown for certain multivariate distributions. Applications are presented for the agricultural industry using USDA database, as well as a financial portfolio problem using recent US stock market data.