In a medium saucepan, cover the potatoes with water and bring to a boil. Add a generous pinch of salt and simmer over moderate heat until tender, about 15 minutes. Drain and spread the potatoes on a plate to cool slightly, then peel and cut into 1/3-inch-thick slices. Transfer the potatoes to a medium bowl and toss with 2 tablespoons of the olive oil; season generously with salt and pepper.
Meanwhile, in a large skillet, heat the remaining 2 tablespoons of olive oil. Add the radicchio and cook over moderately high heat, tossing, until just wilted, about 3 minutes. Stir in the vinegar and season with salt and pepper.
Arrange the sliced potatoes on a platter or plates and spoon the radicchio on top. Scatter the cheese and apple over the salad, then finely grate fresh horseradish on top. Serve, passing additional freshly grated horseradish at the table.