In a low-statistics PET imaging context, the positive bias in regions of low activity is a burning issue. To overcome this problem, algorithms without the built-in non-negativity constraint may be used. They allow negative voxels in the image to reduce, or even to cancel the bias. However, such algorithms increase the variance and are difficult to interpret since the resulting images contain negative activities, which do not hold a physical meaning when dealing with radioactive concentration. In this paper, a post-processing approach is proposed to remove these negative values while preserving the local mean activities. Its original idea is to transfer the value of each voxel with negative activity to its direct neighbors under the constraint of preserving the local means of the image. In that respect, the proposed approach is formalized as a linear programming problem with a specific symmetric structure, which makes it solvable in a very efficient way by a dual-simplex-like iterative algorithm. The relevance of the proposed approach is discussed on simulated and on experimental data. Acquired data from an yttrium-90 phantom show that on images produced by a non-constrained algorithm, a much lower variance in the cold area is obtained after the post-processing step, at the price of a slightly increased bias. More specifically, when compared with the classical OSEM algorithm, images are improved, both in terms of bias and of variance.