The Honeycomb Conjecture states that among tilings with unit area cells in the Euclidean plane, the average perimeter of a cell is minimal for a regular hexagonal tiling. This conjecture goes back to a book of the Roman polyhistor Varro, but it was proved only in the middle of the 20th century by L. Fejes Toth for convex tilings, and in the beginning of the 21th century by Hales for not necessarily convex tilings. It seems a natural question to ask whether in any normed plane, among tilings with unit area cells, the average perimeter is minimal for a tiling whose cells are translates  of a given (not necessarily regular) hexagon. In this talk we investigate this question for convex tilings in normed planes. We show that the answer is affirmative in any normed plane for tilings with minimal average squared perimeter, and show that the original problem in general is related to an \alpha-convex variant of Dowker's theorem on the areas of polygons circumscribed about a plane convex body. Exploring this connection, we show that the same statement with average perimeter holds for any norm whose unit disk is a regular (2n)-gon with n not equal to 4,5,7. This is an joint work with Zsolt Langi.