Place all the cut up vegetables in a large bowl, toss to mix up.
In a med size bowl, whisk together vinegar, oil, reserved fresh tomato juice, canned tomato juice, and set aside.
In a food processor fitted with the steel blade, in batches, pulse the mixed vegetables until the pieces are pea sized. Empty the processor after each batch is pulsed - add to an empty bowl. After all the veggies are pulsed, begin adding them back to the processor to even all the vegetable to the same consistency, and purée a little more this time. Add a little of the tomato juice mixture each time to the processor. Do the final mix in your large bowl. You should be able to see the pieces but not really tell what they are. The gazpacho should be pureed but not to a tomato juice consistency; it should still have a crunchy texture.
Stir in the cayenne, salt, pepper and dill. Add more tomato juice if the soup needs thinning.
Cover and chill for at least 4 hours; the flavors become more pronounced as it sits (even better...serve it a day or two later).
his gazpacho is best made at least one day ahead. It will last 1 week in the refrigerator; it gets better with age.