This paper presents a mathematical methodology to derive optimal margin and edge-enhanced intensity maps in radiation therapy, aiming to provide tumor coverage while minimizing the total dose delivered. The authors analyze the impact of motion and uncertainty on these intensity maps and provide formulas to calculate their dimensions. Key findings include: 1. **Optimal Margins**: If the tumor size is less than or equal to 2.28 times the standard deviation of motion, the optimal intensity map is a pure scaling increase of the static intensity map without any margins or edge-enhancements. If the tumor size is greater than 2.28 times the standard deviation of motion, an isotropic margin around the tumor is optimal, and a non-linear equation is derived to determine the optimal margin size. 2. **Robustness to Motion Uncertainty**: The paper extends the analysis to scenarios where the parameters of the motion distribution are uncertain but lie within certain ranges. The optimal margin size is determined by a modified ratio of the effective tumor size divided by the maximum standard deviation, with a threshold of approximately 2.28. 3. **Edge-Enhanced Intensity Maps**: These maps are characterized by areas of high intensity at the tumor's edge, coupled with lower, uniform intensity in the interior. If the ratio of the tumor size to the standard deviation of motion is less than 2.11, the optimal intensity map is a pure scaling increase. If the ratio is greater than 2.11, edge-enhancements are optimal, and equations are derived to determine their height and width. The paper provides a comprehensive framework for determining optimal intensity maps in the presence of motion and uncertainty, offering practical guidelines for treatment planning.This paper presents a mathematical methodology to derive optimal margin and edge-enhanced intensity maps in radiation therapy, aiming to provide tumor coverage while minimizing the total dose delivered. The authors analyze the impact of motion and uncertainty on these intensity maps and provide formulas to calculate their dimensions. Key findings include: 1. **Optimal Margins**: If the tumor size is less than or equal to 2.28 times the standard deviation of motion, the optimal intensity map is a pure scaling increase of the static intensity map without any margins or edge-enhancements. If the tumor size is greater than 2.28 times the standard deviation of motion, an isotropic margin around the tumor is optimal, and a non-linear equation is derived to determine the optimal margin size. 2. **Robustness to Motion Uncertainty**: The paper extends the analysis to scenarios where the parameters of the motion distribution are uncertain but lie within certain ranges. The optimal margin size is determined by a modified ratio of the effective tumor size divided by the maximum standard deviation, with a threshold of approximately 2.28. 3. **Edge-Enhanced Intensity Maps**: These maps are characterized by areas of high intensity at the tumor's edge, coupled with lower, uniform intensity in the interior. If the ratio of the tumor size to the standard deviation of motion is less than 2.11, the optimal intensity map is a pure scaling increase. If the ratio is greater than 2.11, edge-enhancements are optimal, and equations are derived to determine their height and width. The paper provides a comprehensive framework for determining optimal intensity maps in the presence of motion and uncertainty, offering practical guidelines for treatment planning.