Add the potatoes to a stockpot and cover with two inches of water. Bring to a boil over medium-high heat. Once boiling, reduce the heat to a gentle boil and continue to cook the potatoes until a knife can be easily inserted through their centers, but they are still firm, approximately 10 minutes. Drain and set aside.
In a medium bowl, mix together the yogurt, mayonnaise, lemon juice, dill, salt and pepper. Cut the apple into ¼-inch-wide matchsticks and add to the bowl. Toss gently to combine.
Slice the potatoes crosswise into ½-inch rounds and arrange on a serving platter. Top with the sliced radishes and then with the dressed apple sticks. Finish with more freshly ground black pepper, if desired. Serve immediately or keep refrigerated for up to 1 hour before serving.