Results Math (2019) 74:13 Online First c  2018 Springer Nature Switzerland AG https://doi.org/10.1007/s00025-018-0938-y Results in Mathematics An Answer to an Open Problem on the Multivariate Bernstein Polynomials on a Simplex Ioan Gavrea and Mircea Ivan Dedicated to Professor Heiner Gonska on the occasion of his 70th birthday Abstract. We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle. Mathematics Subject Classification. 32E20, 41A36, 41A17, 42A16. Keywords. Multivariate Bernstein polynomials, simplex, convexity. 1. The Open Problem There is an increasing interest in studying the properties of the sums of squared Bernstein polynomials (see, e.g., [1, 3–6] and many papers in http://arxiv.org/). In this paper, we focus on a very recent open problem on the subject. Let T 2 := {(x, y) ∈ R 2 | x ≥ 0,y ≥ 0,x + y ≤ 1} and n be a positive integer. For (x, y) ∈ T 2 , A 2,n (x, y) :=  i+j≤n  n! i!j !(n − i − j )! x i y j (1 − x − y) n-i-j  2 , is the sum of squares of the functions from the Bernstein basis on the unit triangle T 2 . The following open problem concerning A 2,n was recently posed. Problem 1 [7, Problem 1]. Is A 2,n convex on T 2 ? 0123456789().: V,-vol