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Fix natural numbers \(m,n\) and \(r\), and consider real-valued functions defined on the unit interval. Such a function is \(m\)-convex if its \(m\)th derivative exists and is positive everywhere. Let \({\mathcal P}_{m,n}\) denote the set of all projections from the space of \(r\) times continuously differentiable functions onto the subspace of \(n\)th degree polynomials with the property that each \(m\)-convex function is mapped to an \(m\)-convex polynomial. (The authors suppress the dependence on \(r\).) This paper is predominantly concerned with the case \(m=n-1\). Under the assumption \(2\leq n\leq r+1\), it is shown that \({\mathcal P}_{n-1,n}\) is an affine subspace, each element of which has norm at least \(3\over 2\). The existence of an element with norm exactly \(3\over 2\), which is unique only for \(n=2\), is established by an explicit formula. In the case \(r=0\), the authors recall a minimal element of \({\mathcal P}_{3,2}\), which has norm less than \(5\over 4\), and show that it belongs to \({\mathcal P}_{2,2}\).