I will admit that they have a point about the link between low carb diets and low blood pressure. I have experienced low blood pressure frequently while on Atkins. However, in many ways recognizing this link kind of proves the merrits of Atkins.

High blood pressure is greatly exacerbated by, if not caused by, insulin. Insulin causes retention of fluids and salt. On atkins, fluids are lost more rapidly due to low insulin levels, and therefore minerals also lost more rapidly. This results in low electrolyte levels and that feeling of dizziness when chaning position from lying down to sitting. It has nothing whatsoever to do with carbs and everything to do with low minerals. Supplementing with more potassium based salt (and sometimes, regular sodium) will usually make this go away.

Or, eating the juices from steak, and choosing high mineral low carb veggies like spinach and avocado is also a good idea.

Ironically, the reason starchy and sugary veggies tend to be higher in minerals is because they have so much energy stored in them; they are fat. Vegitation stores energy using insulin just as animals like humans do, and so in the veggies that also tend to be loaded with "fat" like starch and sugar, also tend to be clinging on to minerals.