We present a deterministic polynomial-time algorithm for estimating the volume of a hypercube intersected by a fixed number of constraints, where each constraint is the sum of univariate functions that are each nonnegative, monotone, and convex. Such constraints include knapsack and norm-ball constraints. The case of the unit hypercube truncated by a single linear constraint (halfspace) is already #P-hard. Given a fixed number of such constraints in a given dimension, together with an error parameter, our algorithm computes a multiplicative approximation of the volume of their intersection with the unit hypercube in time polynomial in the dimension, reciprocal error, input encoding length, and output encoding length.

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